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Trend analysis of monthly acid rain data - '80 -'86 Wu, Shiying 1988

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T R E N D ANALYSIS OF M O N T H L Y ACID RAIN DATA By SHIYING W U  B. S. (Mathematics), Peking University, 1983  A THESIS SUBMITTED IN PARTIAL F U L F I L L M E N T O F T H E REQUIREMENTS F O R T H E D E G R E E O F M A S T E R OF SCIENCE  in T H E F A C U L T Y OF G R A D U A T E STUDIES D E P A R T M E N T OF STATISTICS  We accept this thesis as conforming to the required standard  T H E UNIVERSITY O F BRITISH COLUMBIA  October 1988  ©  SHIYING WU,  1988  In  presenting  degree freely  at  this  the  available  copying  of  department publication  of  in  partial  fulfilment  University  of  British  Columbia,  for  this or  thesis  reference  thesis by  this  for  his  and  scholarly  or  thesis  for  her  of  The University of British Vancouver, Canada  Da,e  DE-6 (2/88)  Columbia  ft,* | 7,.  I further  purposes  gain  the  requirements  I agree  shall  that  agree  may  representatives.  financial  permission.  Department  study.  of  be  It not  is  that  the  Library  permission  granted  by  understood be  for  allowed  an  advanced  shall for  the that  without  make  it  extensive  head  of  my  copying  or  my  written  ABSTRACT  Three-way median polish is used to model the monthly concentrations of three kinds of ions in precipitation, namely sulphate, nitrate and hydrogen ions. In contrast to previous findings that the wet acid deposition had decreased from late 70's to early 80's, the results suggest that there is a V-shaped trend for wet acid deposition during the period of 1986 with the change point around 1983. analysis.  1980 -  Strong seasonality is also discovered by the  Nonparametric monotone trend tests are performed on the data collected from  1980 to 1986 and on the data collected from 1983 to 1986 separately. consistent with the findings from the median polish approach. estimate of the trend is obtained for each monitoring station.  The results are  A nonparametric slope  Based on these estimates,  the slope estimate is obtained by Kriging interpolation for each integer degree grid point of longitude and latitude across the 48 conterminous states in the United States. Also, a geographical pattern in the data is suggested by hierarchical clustering and by median polishing.  T A B L E O F CONTENTS  ABSTRACT  ii  T A B L E OF CONTENTS  iii  LIST OF T A B L E S  v  LIST O F FIGURES  vii  ACKNOWLEDEMENT  xx  CHAPTER  1  INTRODUCTION  1  1.1  Purpose and scope  1  1.2  A Review of Previous Works  2  1.3  Data Base Description  3  CHAPTER 2  METHODOLOGY BACKGROUND  6  2.1  Clustering  6  2.2  Nonparametric Monotone Trend Test and Slope Estimator  8  2.3  Median Polish  16  2.4  Kriging and Universal Kriging  21  CHAPTER 3  APPLICATIONS A N D O V E R V I E W  32  3.1  Transformation and Clustering  32  3.2  Trend and Seasonality  33  3.3  Nonparametric Test, Slope Estimate and Kriging  34  CHAPTER 4 4.1  4.2  A N A L Y S E S A N D CONCLUSIONS FOR S U L P H A T E  Results for the Historical Data  37 37  4.1.1  Clustering and Transformation  37  4.1.2  Trend, Seasonality and Spatial Patterns  38  4.1.3  The Results of Trend Testing, Slope Estimation and Kriging  40  Results for the Recent Data 4.2.1  42  Clustering and Transformation  i ii  42  4.2.2  Trend, Seasonality and Spatial Patterns  43  4.2.3  The Results of Trend Testing, Slope Estimation and Kriging  45  CHAPTER 5 5.1  ) 5.2  6.2  Ill  Results for the Historical Data  Ill  5.1.1  Clustering and Transformation  Ill  5.1.2  Trend, Seasonality and Spatial Patterns  112  5.1.3  The Results of Trend Testing, Slope Estimation and Kriging  114  Results for the Recent Data  116  5.2.1  Clustering and Transformation  116  5.2.2  Trend, Seasonality and Spatial Patterns  117  5.2.3  The Results of Trend Testing, Slope Estimation and Kriging  119  ANALYSES AND CONCLUSIONS FOR HYDROGEN ION  180  CHAPTER 6 6.1  ANALYSES AND CONCLUSIONS FOR NITRATE  Results for the Historical Data  180  6.1.1  Clustering and Transformation  180  6.1.2  Trend, Seasonality and Spatial Patterns  181  6.1.3  The Results of Trend Testing, Slope Estimation and Kriging  183  Results for the Recent Data  .'  6.2.1  Clustering and Transformation  6.2.2  Trend, Seasonality and Spatial Patterns  6.2.3  The Results of Trend Testing, Slope Estimation and Kriging  CHAPTER 7  SUMMARY AND FURTHER STUDIES  BIBLIOGRAPHY  184 185  .'  185 187 251 253  i  V  LIST OF T A B L E S  TABLE 4.1.1  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Sulphate (Monthly Volume Weighted Mean, '80-'86)  4.1.2  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Sulphate (Monthly Median, '80-'86)  4.2.1  176  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Nitrate (Monthly Median, '83-'86)  6.1.1  175  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Nitrate (Monthly Volume Weighted Mean, '83-'86)  5.2.2  174  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Nitrate (Monthly Median, '80-'86)  5.2.1  109  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Nitrate (Monthly Volume Weighted Mean, '80-'86)  5.1.2  107  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Sulphate (Monthly Median, '83-'86)  5.1.1  106  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Sulphate (Monthly Volume Weighted Mean, '83-'86)  4.2.2  105  178  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Hydrogen ion (Monthly Volume Weighted Mean, '80-'86)  6.1.2  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Hydrogen ion (Monthly Median, '80-'86)  6.2.1  245  246  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Hydrogen ion (Monthly Volume Weighted Mean, '83-'86)  v  247  6.2.2  Results of the Mann-Kendall Tests and Slope Estimats for the Concentrations of Hydrogen ion (Monthly Median, '83-'86)  v i  249  LIST OF FIGURES  FIGURE 4.1.1  The Location of the 31 Monitoring Stations from 1980 to 1986 (For Sulphate)  4.1.2 (a)  47  Clustering of S04 Monthly Volume Weighted Mean based on sqrt(MSE), 1980 - 1986  4.1.2 (b)  Clustering of S04 Monthly Volume Weighted Mean based on sqrt(MSE), 1980 - 1986 (with outliers)  4.1.3  48  49  Clusters of S04 Monthly Volume Weighted Mean based on sqrt(MSE), 1980 - 1986 (k=3)  50  4.1.4  Histograms of Transformed S04 (Volume Weighted Mean, 80-86)  51  4.1.5  Histograms of log(S04) by Clusters (Volume Weighted Mean, 80-86) .. 52  4.1.6  Yearly Effect of log(S04) for 3 Clusters (monthly volume weighted mean, 1980-1986)  4.1.7  Yearly Effect of log(S04) for 3 Clusters (monthly median, 1980-1986)  4.1.8  57  Station Effect of log(S04) for Cluster 2 (monthly median/volume weighted mean, 1980-1986)  4.1.12  56  Station Effect of log(S04) for Cluster 1 (monthly median/volume weighted mean, 1980-1986)  4.1.11  55  Monthly Effect of log(S04) for 3 Clusters (monthly median, 1980-1986)  4.1.10  54  Monthly Effect of log(S04) for 3 Clusters (monthly volume weighted mean, 1980-1986)  4.1.9  53  58  Station Effect of log(S04) for Cluster 3 (monthly median/volume weighted mean, 1980-1986) v i i  59  4.1.13 (a) Boxplot for the Resid. of log(S04) (80-86, monthly-volume weighted mean, clust 1)  60  4.1.13 (b) Boxplot for the Resid. of log(S04) (80-86, monthly median, clust 1)  61  4.1.14 (a) Boxplot for the Resid. of log(S04) (80-86, monthly volume weighted mean, clust 2)  62  4.1.14 (b) Boxplot for the Resid. of log(S04) (80-86, monthly median, clust 2)  63  4.1.15 (a) Boxplot for the Resid. of log(S04) (80-86, monthly volume weighted mean, clust 3)  64  4.1.15 (b) Boxplot for the Resid. of log(S04) (80-86, monthly median, clust 3)  65  4.1.16 (a) Summary of the Effects and Residuals from Median Polish of log(S04) (80-86, monthly volume weighted mean, clust 1)  66  4.1.16 (b) Summary of the Effects and Residuals from Median Polish of log(S04) (80-86, monthly median, clust 1)  67  4.1.17 (a) Summary of the Effects and Residuals from Median Polish of log(S04) (80-86, monthly volume weighted mean, clust 2)  68  4.1.17 (b) Summary of the Effects and Residuals from Median Polish of log(S04) (80-86, monthly median, clust 2)  69  4.1.18 (a) Summary of the Effects and Residuals from Median Polish of log(S04) (80-86, monthly volume weighted mean, clust 3)  70  4.1.18 (b) Summary of the Effects and Residuals from Median Polish of log(S04) (80-86, monthly median, clust 3)  71  4.1.19 (a) Trend of log(S04) from 1980 to 1986 at the 31 Stations (calculated by monthly volume weighted mean)  viii  72  4.1.19 (b) Trend of log(S04) from 1980 to 1986 at the 31 Stations (calculated by monthly median)  73  4.1.20 (a) Trend of log(S04) from 1980 to 1986 in the U S A (calculated by Kriging from monthly volume weighted mean) 4.1.20  (b) Trend of log(S04) from 1980 to 1986 in the U S A (calculated by Kriging from monthly median)  4.2.1  4.2.2 (a)  76  Clustering of S04 Monthly Volume Weighted Mean based on sqrt(MSE), 1983 - 1986  4.2.2 (b)  75  The Location of the 81 Monitoring Stations from 1983 to 1986 (For Sulphate)  77  Clustering of S04 Monthly Volume Weighted Mean based on sqrt(MSE), 1983 - 1986 (with outliers)  4.2.3  74  78  Clusters of S04 Monthly Volume Weighted Mean based on sqrt(MSE), 1983 - 1986 (k=3)  79  4.2.4  Histograms of Transformed S04 (Volume Weighted Mean, 83-86)  4.2.5  Histograms of log(S04) by Clusters (Volume Weighted Mean, 83-86) .. 81  4.2.6  Yearly Effect of log(S04) for 3 Clusters (monthly volume weighted mean, 1983-1986)  4.2.7  84  Monthly Effect of log(S04) for 3 Clusters (monthly median, 1983-1986)  4.2.10  83  Monthly Effect of log(S04) for 3 Clusters (monthly volume weighted mean, 1983-1986)  4.2.9  82  Yearly Effect of log(S04) for 3 Clusters (monthly median, 1983-1986)  4.2.8  80  85  Station Effect of log(S04) for Cluster 1 (monthly median/volume weighted mean, 1983-1986)  ix  86  4.2.11  Station Effect of log(S04) for Cluster 2 (monthly median/volume weighted mean, 1983-1986)  4.2.12  87  Station Effect of log(S04) for Cluster 3 (monthly median/volume weighted mean, 1983-1986)  88  4.2.13 (a) Boxplot for the Resid. of log(S04) (83-86, monthly volume weighted mean, clust 1)  89  4.2.13 (b) Boxplot for the Resid. of log(S04) (83-86, monthly median, clust 1)  90  4.2.14 (a) Boxplot for the Resid. of log(S04) (83-86, monthly volume weighted mean, clust 2)  91  4.2.14 (b) Boxplot for the Resid. of log(S04) (83-86, monthly median, clust 2)  92  4.2.15 (a) Boxplot for the Resid. of log(S04) (83-86, monthly volume weighted mean, clust 3)  93  4.2.15 (b) Boxplot for the Resid. of log(S04) (83-86, monthly median, clust 3)  94  4.2.16 (a) Summary of the Effects and Residuals from Median Polish of log(S04) (83-86, monthly volume weighted mean, clust 1)  95  4.2.16 (b) Summary of the Effects and Residuals from Median Polish of log(S04) (83-86, monthly median, clust 1)  96  4.2.17 (a) Summary of the Effects and Residuals from Median Polish of log(S04) (83-86, monthly volume weighted mean, clust 2)  97  4.2.17 (b) Summary of the Effects and Residuals from Median Polish of log(S04) (83-86, monthly median, clust 2)  98  4.2.18 (a) Summary of the Effects and Residuals from Median Polish of log(S04) (83-86, monthly volume weighted mean, clust 3)  x  99  4.2.18 (b) Summary of the Effects and Residuals from Median Polish of log(S04) (83-86, monthly median, clust 3)  100  4.2.19 (a) Trend of log(S04) from 1983 to 1986 at the 81 Stations (calculated by monthly volume weighted mean)  101  4.2.19 (b) Trend of log(S04) from 1983 to 1986 at the 81 Stations (calculated by monthly median)  102  4.2.20 (a) Trend of log(S04) from 1983 to 1986 in the U S A (calculated by Kriging from monthly volume weighted mean)  103  4.2.20 (b) Trend of log(S04) from 1983 to 1986 in the U S A (calculated by Kriging from monthly median) 5.1.1  The Location of the 31 Monitoring Stations from 1980 to 1986 (For Nitrate)  5.1.2 (a)  121  Clustering of N03 Monthly Volume Weighted Mean based on sqrt(MSE), 1980 - 1986  5.1.2 (b)  122  Clustering of N03 Monthly Volume Weighted Mean based on sqrt(MSE), 1980 - 1986 (with outliers)  5.1.3  104  123  Clusters of N03 Monthly Volume Weighted Mean based on sqrt(MSE), 1980 - 1986 (k=2)  124  5.1.4  Histograms of Transformed N03 (Volume Weighted Mean, 80-86)  125  5.1.5  Histograms of log(N03) by Clusters (Volume Weighted Mean, 80-86)..126  5.1.6  Yearly Effect of log(N03) for 2 Clusters (monthly volume weighted mean, 1980-1986)  5.1.7  Yearly Effect of log(N03) for 2 Clusters (monthly median, 1980-1986)  5.1.8  127  128  Monthly Effect of log(N03) for 2 Clusters (monthly volume weighted mean, 1980-1986)  xi  129  5.1.9  Monthly Effect of log(N03) for 2 Clusters (monthly median, 1980-1986)  5.1.10  Station Effect of log(N03) for Cluster 1 (monthly median/volume weighted mean, 1980-1986)  5.1.11  130  131  Station Effect of log(N03) for Cluster 2 (monthly median/volume weighted mean, 1980-1986)  132  5.1.12 (a) Boxplot for the Resid. of log(N03) (80-86, monthly volume weighted mean, clust 1)  133  5.1.12 (b) Boxplot for the Resid. of log(N03) (80-86, monthly median, clust 1)  134  5.1.13 (a) Boxplot for the Resid. of log(N03) (80-86, monthly volume weighted mean, clust 2)  135  5.1.13 (b) Boxplot for the Resid. of log(N03) (80-86, monthly median, clust 2)  136  5.1.14 (a) Summary of the Effects and Residuals from Median Polish of log(N03) (80-86, monthly volume weighted mean, clust 1)  137  5.1.14 (b) Summary of the Effects and Residuals from Median Polish of log(N03) (80-86, monthly median, clust 1)  138  5.1.15 (a) Summary of the Effects and Residuals from Median Polish of log(N03) (80-86, monthly volume weighted mean, clust 2)  139  5.1.15 (b) Summary of the Effects and Residuals from Median Polish of log(N03) (80-86, monthly median, clust 2)  140  5.1.16 (a) Trend of log(N03) from 1980 to 1986 at the 31 Stations (calculated by monthly volume weighted mean)  141  5.1.16 (b) Trend of log(N03) from 1980 to 1986 at the 31 Stations (calculated by monthly median)  x ii  142  5.1.17 (a) Trend of log(N03) from 1980 to 1986 in the U S A (calculated by Kriging from monthly volume weighted mean)  143  5.1.17 (b) Trend of log(N03) from 1980 to 1986 in the U S A (calculated by Kriging from monthly median) 5.2.1  The Location of the 81 Monitoring Stations from 1983 to 1986 (For Nitrate)  5.2.2 (a)  145  Clustering of N03 Monthly Volume Weighted Mean based on sqrt(MSE), 1983 - 1986  5.2.2 (b)  146  Clustering of N03 Monthly Volume Weighted Mean based on sqrt(MSE), 1983 - 1986 (with outliers)  5.2.3  144  147  Clusters of N03 Monthly Volume Weighted Mean based on sqrt(MSE), 1983 - 1986 (k=3)  148  5.2.4  Histograms of Transformed N 0 3 (Volume Weighted Mean, 83-86)  149  5.2.5  Histograms of log(N03) by Clusters (Volume Weighted Mean, 83-86). 150  5.2.6  Yearly Effect of log(N03) for 3 Clusters (monthly volume weighted mean, 1983-1986)  5.2.7  Yearly Effect of log(N03) for 3 Clusters (monthly median, 1983-1986)  5.2.8  154  Station Effect of log(N03) for Cluster 1 (monthly median/volume weighted.mean, 1983-1986)  5.2.11  153  Monthly Effect of log(N03) for 3 Clusters (monthly median, 1983-1986)  5.2.10  152  Monthly Effect of log(N03) for 3 Clusters (monthly volume weighted mean, 1983-1986)  5.2.9  151  155  Station Effect of log(N03) for Cluster 2 (monthly median/volume weighted mean, 1983-1986)  xiii  156  5.2.12  Station Effect o f log(N03) for Cluster 3 (monthly median/volume weighted mean, 1983-1986)  157  5.2.13 (a) Boxplot for the Resid. o f log(N03) (83-86, monthly volume weighted mean, clust 1)  158  5.2.13 (b) Boxplot for the Resid. oflog(N03) (83-86, monthly median, clust 1)  :  159  5.2.14 (a) Boxplot for the Resid. o f log(N03) (83-86, monthly volume weighted mean, clust 2)  160  5.2.14 (b) Boxplot for the Resid. o f log(N03) (83-86, monthly median, clust 2)  161  5.2.15 (a) Boxplot for the Resid. o f log(N03) (83-86, monthly volume weighted mean, clust 3)  162  5.2.15 (b) Boxplot for the Resid. o f log(N03) (83-86, monthly median, clust 3)  163  5.2.16 (a) Summary o f the Effects and Residuals from Median Polish o f log(N03) (83-86, monthly volume weighted mean, clust 1)  164  5.2.16 (b) Summary of the Effects and Residuals from Median Polish o f log(N03) (83-86, monthly median, clust 1)  165  5.2.17 (a) Summary o f the Effects and Residuals from Median Polish o f log(N03) (83-86, monthly volume weighted mean, clust 2)  166  5.2.17 (b) Summary o f the Effects and Residuals from Median Polish o f log(N03) (83-86, monthly median, clust 2)  167  5.2.18 (a) Summary of the Effects and Residuals from Median Polish of log(N03) (83-86, monthly volume weighted mean, clust 3)  168  5.2.18 (b) Summary of the Effects and Residuals from Median Polish o f log(N03) (83-86, monthly median, clust 3)  xiv  169  5.2.19 (a) Trend of log(N03) from 1983 to 1986 at the 81 Stations (calculated by monthly volume weighted mean)  170  5.2.19 (b) Trend of log(N03) from 1983 to 1986 at the 81 Stations (calculated by monthly median)  171  5.2.20 (a) Trend of log(N03) from 1983 to 1986 in the U S A (calculated by Kriging from monthly volume weighted mean)  172  5.2.20 (b) Trend of log(N03) from 1983 to 1986 in the U S A (calculated by Kriging from monthly median) 6.1.1  The Location of the 32 Monitoring Stations from 1980 to 1986 (For Hydrogen ion)  6.1.2  189  Clustering of H+ Monthly Volume Weighted Mean based on sqrt(MSE), 1980 - 1986  6.1.3  173  190  Clusters of H+ Monthly Volume Weighted Mean based on sqrt(MSE), 1980 - 1986 (k=3)  191  6.1.4  Histograms of Transformed H+ (Volume Weighted Mean, 80-86)  192  6.1.5  Histograms of log(H+) by Clusters (Volume Weighted Mean, 80-86) .. 193  6.1.6  Yearly Effect of log(H+) for 3 Clusters (monthly volume weighted mean, 1980-1986)  6.1.7  Yearly Effect of log(H+) for 3 Clusters (monthly median, 1980-1986)  6.1.8  197  Station Effect of log(H+) for Cluster 1 (monthly median/volume weighted mean, 1980-1986)  6.1.11  196  Monthly Effect of log(H+) for 3 Clusters (monthly median, 1980-1986)  6.1.10  195  Monthly Effect of log(H+) for 3 Clusters (monthly volume weighted mean, 1980-1986)  6.1.9  194  Station Effect of log(H+) for Cluster 2  xv  198  (monthly median/volume weighted mean, 1980-1986) 6.1.12  199  Station Effect of log(H+) for Cluster 3 (monthly median/volume weighted mean, 1980-1986)  200  6.1.13 (a) Boxplot for the Resid. of log(H+) (80-86, monthly volume weighted mean, clust 1)  201  6.1.13 (b) Boxplot for the Resid. of log(H+) (80-86, monthly median, clust 1)  202  6.1.14 (a) Boxplot for the Resid. of log(H+) (80-86, monthly volume weighted mean, clust 2)  203  6.1.14 (b) Boxplot for the Resid. of log(H+) (80-86, monthly median, clust 2)  204  6.1.15 (a) Boxplot for the Resid. of log(H+) (80-86, monthly volume weighted mean, clust 3)  205  6.1.15 (b) Boxplot for the Resid. of log(H+) (80-86, monthly median, clust 3)  206  6.1.16 (a) Summary of the Effects and Residuals from Median Polish of log(H+) (80-86, monthly volume weighted mean, clust 1)  207  6.1.16 (b) Summary of the Effects and Residuals from Median Polish of log(H+) (80-86, monthly median, clust 1)  208  6.1.17 (a) Summary of the Effects and Residuals from Median Polish of log(H+) (80-86, monthly volume weighted mean, clust 2)  209  6.1.17 (b) Summary of the Effects and Residuals from Median Polish of log(H+) (80-86, monthly median, clust 2)  210  6.1.18 (a) Summary of the Effects and Residuals from Median Polish of log(H+) (80-86, monthly volume weighted mean, clust 3)  211  6.1.18 (b) Summary of the Effects and Residuals from Median Polish of log(H+) (80-86, monthly median, clust 3) xvi  212  6.1.19 (a) Trend of log(H+) from 1980 to 1986 at the 32 Stations (calculated by monthly volume weighted mean)  213  6.1.19 (b) Trend of log(H+) from 1980 to 1986 at the 32 Stations (calculated by monthly median)  214  6.1.20 (a) Trend of log(H+) from 1980 to 1986 in the U S A (calculated by Kriging from monthly volume weighted mean)  215  6.1.20 (b) Trend of log(H+) from 1980 to 1986 in the U S A (calculated by Kriging from monthly median) 6.2.1  The Location of the 86 Monitoring Stations from 1983 to 1986 (For Hydrogen ion)  6.2.2  217  Clustering of H+ Monthly Volume Weighted Mean based on sqrt(MSE), 1983 - 1986  6.2.3  216  218  Clusters of H+ Monthly Volume Weighted Mean based on sqrt(MSE), 1983 - 1986 (k=3)  219  6.2.4  Histograms of Transformed H+ (Volume Weighted Mean, 83-86)  220  6.2.5  Histograms of log(H-t-) by Clusters (Volume Weighted Mean, 83-86)... 221  6.2.6  Yearly Effect of log(H+) for 3 Clusters (monthly volume weighted mean, 1983-1986)  6.2.7  Yearly Effect of log(H+) for 3 Clusters (monthly median, 1983-1986)  6.2.8  224  Monthly Effect of log(H+) for 3 Clusters (monthly median, 1983-1986)  6.2.10  223  Monthly Effect of log(H+) for 3 Clusters (monthly volume weighted mean, 1983-1986)  6.2.9  222  225  Station Effect of log(H+) for Cluster 1 (monthly median/volume weighted mean, 1983-1986)  xvii  226  6.2.11  Station Effect of log(H+) for Cluster 2 (monthly median/volume weighted mean, 1983-1986)  6.2.12  227  Station Effect of log(H+) for Cluster 3 (monthly median/volume weighted mean, 1983-1986)  .*  228  6.2.13 (a) Boxplot for the Resid. of log(H+) (83-86, monthly volume weighted mean, clust 1)  229  6.2.13 (b) Boxplot for the Resid. of log(H+) (83-86, monthly median, clust 1)  230  6.2.14 (a) Boxplot for the Resid. of log(H+) (83-86, monthly volume weighted mean, clust 2)  231  6.2.14 (b) Boxplot for the Resid. of log(H+) (83-86, monthly median, clust 2)  232  6.2.15 (a) Boxplot for the Resid. of log(H+) (83-86, monthly volume weighted mean, clust 3)  233  6.2.15 (b) Boxplot for the Resid. of log(H+) (83-86, monthly median, clust 3)  234  6.2.16 (a) Summary of the Effects and Residuals from Median Polish of log(H+) (83-86, monthly volume weighted mean, clust 1)  235  6.2.16 (b) Summary of the Effects and Residuals from Median Polish of log(H+) (83-86, monthly median, clust 1)  236  6.2.17 (a) Summary of the Effects and Residuals from Median Polish of log(H+) (83-86, monthly volume weighted mean, clust 2)  237  6.2.17 (b) Summary of the Effects and Residuals from Median Polish of log(H+) (83-86, monthly median, clust 2)  238  6.2.18 (a) Summary of the Effects and Residuals from Median Polish of log(H+) (83-86, monthly volume weighted mean, clust 3)  xviii  239  6.2.18 (b) Summary of the Effects and Residuals from Median Polish of log(H+) (83-86, monthly median, clust 3)  240  6.2.19 (a) Trend of log(H+) from 1983 to 1986 at the 86 Stations (calculated by monthly volume weighted mean)  241  6.2.19 (b) Trend of log(H+) from 1983 to 1986 at the 86 Stations (calculated by monthly median)  242  6.2.20 (a) Trend of log(H+) from 1983 to 1986 in the U S A (calculated by Kriging from monthly volume weighted mean)  243  6.2.20 (b) Trend of log(H+) from 1983 to 1986 in the USA (calculated by Kriging from monthly median)  xix  244  ACKNOWLEDGEMENT  I would like to thank Professor J. V . Zidek for his guidance and assistance in producing this thesis. I would like to express my gratitude to Doctor Jian Liu, who kindly gave me helpful advice and carefully read this thesis. The work reported here relies heavily on the data provided by the Battelle's Pacific Northwest Laboratories; special thanks go to Doctor A . R. Olsen and Judy Sweeney. I also would like to express my appreciation to Doctor H . Joe and P. Schumacher for their assistance with the computing issues. This work was partially supported by the Environmental Protection Agency (U.S.) through a co-operative research agreement with Societal Institute for Mathematics in Society.  XX  Chapter 1  INTRODUCTION  The Acid Deposition System (ADS) is an integrated, centralized repository for data from monitoring networks in North America.  The purposes of A D S are (1) to facilitate  access to deposition data collected by different organizations, (2) to provide annual statistical summaries of available data, and (3) long-term trends.  to maintain the data for assessment of  A complete description of A D S is available in a system design and  user's code manual by Olsen and Slavich (1986). The data set used in this study is obtained from A D S Data Summary (Olsen and Slavich ,1986) and it contains only the data from the N A D P network, the largest and most important network among the networks represented in the A D S Data Base. Three important ions of the data set have been singled out for study in this report: SO4, N O 3 and H  1.1  +  (derived from pH). Separate analyses of each have been done in this study.  Purpose and Scope The primary purpose of this study is to detect and estimate the possible temporal trends  in different levels of chemical constituents of acid deposition at different locations. The analysis is divided into 2 sections, one for the data collected from 1980 to 1986 (the "historical data"), and the other for the data collected from 1983 to 1986 (the "recent data"). In this way we have tried to differentiate between what happened in the past seven years from what happened in the last four years.  1  A secondary purpose of this study is the classification of the stations into groups such that patterns in the deposition chemistry of the stations in each group are similar. The ultimate aim is to reduce the number of stations in the network to reduce the cost of maintaining the network without loosing too much information. But this issue will not be addressed in this report. Spatial patterns and seasonal patterns of the level of chemical concentration in wet deposition are examined as well. Over time, the N A D P network came to include more than 200 monitoring stations. Many pf them either started to operate after 1980 or have a lot of missing data. For convenience and with little apparent loss of information, we consider only the stations with 5 or fewer missing observations.  The resulting sets of stations are different for various  chemicals under consideration. Roughly speaking, there are just over 80 stations providing the recent data and just over 30 stations providing the historical data. The stations are located throughout the United States and there are more stations in the East than in the West.  1.2  A Review of Previous Works A lot of assessments of acid precipitation have been done during the  past years.  Among those are Schertz and Hirsch (1985), Eynon and Switzer (1983), Finklestein (1984), Cape and Fowler (1984), Vong etal (1985), Bilonick (1985, 1987), Le and Petkau (1986), Dana and Easter (1986), Lettenmaier (1986), and Altwicker and Johannes (1987). Finklestein (1984) estimates the  variograms of H , S04, N O 3 and N H 4 . These +  variogram estimates are computed from the observations collected at 31 N A P D stations  2  during the period from July 1979 to June 1980, and are shown to be distance dependent in all cases, increasing with the distance between stations. Bilonick (1985) estimated the space-time semi-variograms of sulfate deposition using the data collected from USGS network and did a 3-dimensional Kriging. He (1987) used indicator Kriging and pointed out that "it is a nonparametric technique" and "is useful for highly skewed data distribution and is resistant to outliers", while "the entire probability distribution function at a given point is estimated". Dana and Easter (1986) applied concentration-time regression analysis to the data from M A P 3 S network, collected from 1977 to 1983, to estimate temporal trends. Their conclusions are that the levels of sulfur, H , N H ^ , and NO^ declined during that period. +  Schertz and Hirsch's study cited above contains three parts: (1) modeling variations in concentration; (2) trend analysis; and (3) multiple station analysis. In the first part, they developed a model: log(C)=a+b*log(P)+c*(27tT)+d*(27tT), and used regression to estimate the parameters.  In the second part, they used the  Seasonal Kendall Test to detect trends in the concentrations of constituents with the conclusion that most of the stations showed no trends or down trends during the period from 1978 to 1983. In the third part, they analyzed the relationship between the correlation coefficients and the distance of pairs of stations. Our study is more closely related to the second part of their work.  1.3  Data Base Description  3  A D S requires that networks provide documentation on their network operation and that a minimum level of information accompany each  sample result.  Networks not able to  provide the required information are unable to transfer data to A D S .  In preparing  concentration and deposition summaries, four related steps occur. First, network protocols and data screening procedures are determined and an algorithm to translate this information along with  the sample results to the A D S data base is constructed.  Second, valid sample  criteria for the data summary are determined. They are: (a) all sampling periods for which it is known that no precipitation occurred are considered valid sample periods; (b) the wet deposition sample must be a wet-only  sample; (c) Wet deposition samples that have  insufficient precipitation to complete a chemical analysis for a specified ion are invalid for that specific ion species; (d) an individual ion species concentration accompanied by a comment code designating the measurement to be "suspect" or "invalid" is declared as an invalid sample; and (e) The actual sampling period for a wet deposition sample must be close to the network's protocol sampling period. each summary are computed. site representativeness  Third, data completeness measures for  Fourth, criteria based on data completeness measures and  are selected for reporting a specific data summary.  For  more  details, see Olsen and Slavich (1986).  One of the networks contained in A D S is N A D P / N T N .  The National  Atmospheric  Deposition Program ( N A D P ) was established in 1978 to monitor trends in the exposure of various ecosystems to acidic deposition in the United States.  The N A D P was created by  the Association of State Agricultural Experiment Stations (originally as North Central Regional Project N C - 1 4 1 , now  Interregional Project IR-7)  to conduct research on  atmospheric deposition and its effects, in cooperation with federal, state, and private research agencies.  A major program objective is to discover and characterize biologically  important geographical and temporal trends in the chemical climate of North America through the  continued development and maintenance of a deposition monitoring network.  4  Since its inception the network has grown from 22 operational sites during 1978 to 222 sites in 1986. The Deposition Monitoring Task Group of the Interagency Task Force on Acid Precipitation was charged in the National Acid Precipitation Assessment Plan (Interagency Task Force on Acid Precipitation, 1982) with developing a National Trends Network (NTN). The objective of the 150-station National Trends Network is to provide long-term monitoring (10-years minimum) at sites across the United States that represent broad regional characteristics of the chemistry of wet deposition. describe the design of N T N .  Robertson and Wilson (1985)  Many existing N A D P sites were selected as N T N sites.  Because of the operating modes of N A D P and N T N , the two networks are considered to be a single network and the acronym N A D P / N T N is used to refer to sites which are either N A D P or both N A D P and NTN. In the following, however, we refer to it simply as NADP. The N A D P monitoring protocol is based on a weekly (Tuesday to Tuesday) sampling protocol with wet-only sample collection. The N A D P program has developed and adheres to strict requirements regarding sample collection and analysis.  The requirements assure  uniformity in siting criteria, sampling protocol, analytical chemistry techniques, data handling, and overall network operation. A l l N A D P precipitation chemistry samples are analyzed by the Central Analytical Laboratory at the Illinois State Water Survey. For specific details the reader may consult existing publications on siting criteria (NADP 1984a), site operation and collection protocol (NADP 1982), overall quality assurance plans (NADP 1984b), and analytical procedures(NADP 1980).  5  Chapter 2  M E T H O D O L O G YBACKGROUND  The statistical procedures involved in this paper are clustering, median polishing, nonparametric trend testing and slope estimation, and Kriging.  2.1  Clustering  In the single sample problem, (x^, —, i = 1,  ) is the ith observed n-dimensional sample, x^,  m, which may well be heterogeneous. The aim of cluster analysis is to group these  samples into g homogeneous classes where g is unknown, g < m. The multi-sample variant involves x^, k = 1, —, n^, the observations of the ith sample x-, i=l, —, m and again the aim is to group the m samples into homogeneous classes (Mardia, Kent, Bibby, 1979).  Here, we consider only hierarchical methods which cluster the g groups into g+1 groups according to the distances between the samples, dividing one group into two and keeping the others the same. One way to do this is follows. Start with Suppose (C^(p), i = 1,  (C-(l) = x-, i = 1,  m}.  m - p + 1} are the clusterings at step p. Define Djj(p) to be a  measure of distance between C-(p) and Cj(p). Let  D ( p ) = min{D .(p):i,j = 1, - , m-p+1, i * j}. 12  ij  Then set CjCp+l) = C ( p ) u C ( p ) 1  2  and C.(p+1) = C  i + 1  (p),  i = 1,  6  m-p+1.  Continue this procedure until all the inter-cluster distances are greater than D Q where D Q is an arbitrary threshold value.  If d  r s  is taken to be the distance between x and x f  g  (the distance can be defined in  different ways), and if  Djj(p) = m i n { d : (r, s) such that x e Cj(p), x e Cj(p)} r s  r  g  the method is called a single linkage method. If Djj(p) = m a x { d : (r, s) such that x € C^p), r s  f  x G C(p)} g  then the method is called a complete linkage method. The  clusters obtained by a single linkage method are "rod" type elongated clusters  without nuclei. This leads to a chaining effect. Chaining can be misleading if items at opposite ends of the chain are quite dissimilar (Johnson and Wichern, 1982). On the other hand, the clusters obtained by the complete linkage method tend to be compact clusters without a chaining effect. Thus the within-group distances of the resulting groups are all less than the threshold value D Q .  Since the ultimate purpose of the clustering used here is to choose representatives from each cluster and we expect the inter-group distances to be small, we choose complete linkage in this study. For the same reason, the distances used in this study are defined as  d  2 ii  n =  X^ik-xjk) ^ - )' k=l 2  11  i, j = 1,  1  where n = n;.  7  m,  I n the s i n g l e s a m p l e case, s u c h a d-j i s e q u i v a l e n t to E u c l i d e a n d i s t a n c e s i n c e  the  m e t h o d i s i n v a r i a n t u n d e r any m o n o t o n i c t r a n s f o r m a t i o n o f d-j. I n m u l t i - s a m p l e p r o b l e m s  2  w h e r e the s a m p l e s are one d i m e n s i o n a l , i f x . has m e a n \X{ a n d v a r i a n c e a- (i = 1,—, m ) , and  2  i f the c o v a r i a n c e o f x . a n d x . i s denoted b y P y , d - i s an estimate o f  2 Var(xj-xj) =  2  G + Oj - 2 Py + {  T h u s a s m a l l d-j r e q u i r e s that b o t h  - pj) ,  i , j = l , - , m.  2  P y be l a r g e a n d that (^j - | i j )  2  be s m a l l . T h i s m e a n s  that the t w o samples v a r y i n a s i m i l a r f a s h i o n .  S o m e p r o b l e m s m a y be c o n s i d e r e d b o t h s i n g l e s a m p l e p r o b l e m a n d m u l t i - s a m p l e p r o b l e m as w e w i l l see i n chapter 3.  2.2  Nonparametric Monotone Trend Test and Slope Estimator  Let  Xp--, x  n  be a s e q u e n c e o f o b s e r v a t i o n o r d e r e d b y t i m e . W e are i n t e r e s t e d i n the  n u l l hypothesis H Q : the o b s e r v a t i o n s are r a n d o m l y ordered, i.e., X p — , X  r  are i . i . d .  samples,  a n d the a l t e r n a t i v e h y p o t h e s i s  H i : there is a m o n o t o n e trend o v e r t i m e , i.e., F  (x) > (or < ) F X  8  j  (x) Xj  for all i < j with at least one strict inequality, where F (x) is the i x  cumulative density function of random variable Xj.  Let f 1 sgn(x) = { 0  x>0 x=0 x<0  I-l  Mann (1945) proposed the following test statistic:  S= X  s  g < j- i>n  x  x  Under H Q , the test statistic has mean 0 and variance  o  2  = [n(n-l)(2n+5)/18]  2  and S/o is asymptotically N(0,1). Kendall (1975) gives the mean and variance of S under H Q given the possibility that there may be ties in the x values:  E(S) = 0 , Var(S) = { n(n-l)(2n+5)-]Tt(t-l)(2t+5) }/18 , t  where t is the extent of any given tie (number of x's involved in a given tie) and ^ denotes t the summation over all ties.  9  Both Mann and Kendall derive the exact distribution of S for n<10 and show that even for n=10 the normal approximation is excellent, provided one uses a continuity correction, i.e., computes the standard normal variate Z by  (S-l)(Var(S))  -1/2  S=0  Z=< 0 ^ (S+l)(Var(S))  S >0  -1/2  S <0  Then in a two-sided test, a positive value of Z indicates an up trend and a negative value of Z indicates a down trend. This test is commonly called Mann-Kendall test. Bradley (1968, p228) notes that when this test is used as a test of randomness against normal regression alternatives, this test has an asymptotic relative efficiency of 0.98 relative to the parametric test based on the regression slope coefficient. Kendall's (1970) x test for correlation considers a more general case. Suppose (Xpyj),---,  ( >y ) are a sequence of bivariate observations, ordered by time. The x  n  n  hypothesis to be tested is:  versus H x*0, i :  where x =2P(X->X.IY.>Y.) - 1. In their procedure , a point estimator of x is given by  10  For  the Mann's test, y- = i, i = 1, —, n.  Sometimes the time series data of interest show that there exist seasonal patterns in the data and thus the hypotheses mentioned above may be too restrictive. However, different procedure are given for dealing with such cases. Let  X = ( X p —, Xp),  where Xg=(x^, —, x g ) is a subsample of season g and x- is the 1  n  a  observation obtained in the ith year and the gth season. A procedure proposed by Dietz and Killen (1981) can be used in this case: The null hypothesis under consideration is that the p vectors are randomly ordered vs. the alternative hypothesis that there is monotone trend in one or mere of the p variables. Let S = (S ,-,S ) , t  1  p  and £ = (a ), gh  where S = £ s g n ( x - x ) > g=l,~,p, g  jg  ig  i<j a =n(n-l)(2n+5)/18, gg  g = 1, - , p,  and  TJX  = °gh  Sgn [ ( x  jg- ig X  [ i<j  ) ( X  Jh* ih X  ) ] +  S  Sgn  (ij.k)  t^jg'V^Jh^kh^l'  J  g * h, the estimated covariances of S and S i . Then S ^ S is asymptotically X , where g n 4 5T is any generalized inverse of X and q<p is the rank of X.  11  Farrell (1980), following Sen (1968), proposed another test procedure in which the data are "deseasonalised" first. Let y-- = x.. - x.. where x.- = — V x . . , R - is the rank of y.. among i all np y 's, 1 = 1, —, n, m = 1, —, p, and H Q  represents the hypothesis of no trend. Then  1m  T = 12p^ n(n + D ^ O R i j -  is asymptotically  R.j) l 2  f^t(  - ( + D/2)(Ri. - (np + l)/2)]  i  n  ~X ij'i- pX ij R  N(0,1), where R j =  1  R  =  R  J  Hirsh et al. (1982) defined a multivariate extension of the Mann-Kendall test called the Seasonal Kendall Test. Let H Q and H j be the hypothesis given by HQ:  (x.y, —, x j) are independent and identical sample for j = 1, —, p and n  x^'s are independent.  and H j : one or more ( iy x  "'•>  x n  j)'  sample. Let  S  g I =  S g n (  i<j  V gi X  ) , g = 1 ,  and  g=l  Then E(S')=0, and  12  '"»  P  s  3  1  6  n  o  t  independent and identical  Var(S')=2^Var(S). j  j=l Let r  (S*-l)(Var(S'))"  Z' = <  S' = 0  (S'+l)(Var(S'))" then Z '  S' >0  1/2  S'<0;  1/2  is asymptotically N(0,1) under H Q .  They demonstrate  that the normal  approximation is quite accurate even for sample sizes as small as n=2, p=12.  The Seasonal Kendall Test is similar to a test proposed by Jonckheere (1954) as a multivariate extension of the sign test for the case when the number of observations is greater than 2. In the case that njS are equal, the seasonal test and Jonckheere's (1954) test are equivalent.  Hirsch and Slack (1984) modified their seasonal Kendall test using Dietz and Killen's estimator of the covariances of S and S, . So the variance of S' becomes g h  Var(S')=^Var(S ) + ^ c o v ( S , S ) . g  g=l  g  h  g*h  The approximation is good in this test when p = 12 and n > 10. Compared with the seasonal Kendall test, this test is less powerful but more robust against serial correlation.  As van Belle and Hughes (1984) suggest in their paper, Hirsch's (1982) and Farrell's (1980) procedures may be based on the same model, i.e. x.. = u + a- +b- + e--, ii I i ii'  i = 1, —, n, j = 1, —, p,  13  where  i  j  is the yearly component, bj is the seasonal component and e^ are i.i.d. with E(e-j) = 0. The common hypothesis being tested is  H  0  :  a  l  =  -  =  a  n  =  0  versus H , : a, < ••• < a or a-. > ••• > a l l n l n with at least one strict inequality. These are true for Hirsch's (1984) modified procedure too. van Belle and Hughes (1984) conclude that Farrell's (1980) procedure is more powerful when there are no missing data since ranking all together preserves the relative ranks between seasons which are lost in the Seasonal Kendall Test, while Hirsch's (1982) is easier to compute when there are missing data. Only the Diets and Killen's test is valid when the trends in different seasons are heterogeneous but Hirsch (1982) argues that it is probably only applicable for sets of data including at least 40 years of monthly values. A l l the other tests of seasonal data mentioned above will be misleading if the trends are not homogeneous among seasons especially when there are opposing trends in different seasons. So van Belle and Hughes (1984) develop a 2-way A N O V A - l i k e nonparametric trend test which can test for the homogeneity of trend direction at different locations and  1  4  different seasons. When there is only one location, it looks like a 1-way A N O V A and the statistic they proposed is  X  2 homog  =  X  2 2 total" trend X  =  P 2 X j " g=l Z  —2 '  Z  where Z  = S (Var(S ))"l/2 ,  g  g  Z=  g  iz /p. g  and S is the Mann - Kendall trend statistic for the gth season, g 6  2 If the trend for each season is in the same direction then ^  n  2 2 distribution with (p-1) degrees of freedom X ^ j . If %  l T c n  o  m  0  g ^  a s a  c  ^  s c  l  u a r e  d  £ exceeds the predefined critical  value, then the null hypothesis of homogeneous seasonal trends is rejected in which case the Seasonal Kendall Test does not apply. However, if that hypothesis is accepted, then 2 ^homog  * ^ s t  i e  s  t  a  u  s  u  c  u s e  d to test the hypothesis that the common trend direction is  significantly different from 0. 2 van Belle and Hughes (1984) point out that the validity of these X the S  tests requires that  be independent. A procedure for testing the contrasts is proposed and possible use  of Newman-Keul's procedure to group the seasons are illustrated in their paper as well. Sen (1968b) developed a nonparametric procedure to estimate the slope of a possible existing trend with a 100(l-oc)% confidence interval and it is robust against gross data errors and outliers. Let  15  Q i j = (Xj -  xp/O - i ) .  Suppose there are N such (i, j) pairs, the median of which is Sen's estimator of the slope. A  simple way to get a 100(1  approximation. Q(m2+1)^ *  ^l-a/2  x % t  n  s a  e  Suppose  QQ)>  ^0(1 "  con  - oc)% confidence interval is by using normal  Q(N)  "'•>  t  ^  n  e  o r <  ^  e r  statistics of the Q j j ' s - Then  [ Q ^  m  j y  fident interval of the slope estimator, where  ml = (N - Z  (Var(S))l/2)/2 ,  m2 = (N+ Z  (Var(S))l/2)/2 ,  " /2) quantile of the standard normal distribution. a  A seasonal Kendall slope estimator is given by Gibert (1987). Suppose there are pairs of x^, i)  such that i < j . Then there are N^. Q^^-values where Q j j ] ^ ( X j ^ -  for the kth season. Let N =  "  — + N^. In all, there are N slope estimates Q - j ^ for all  the seasons conbined. The median of these N Q ^ s is the seasonal Kendall slope estimator and if Q ^ ,  Q.Q$)  3 X 0  ^  o r d e r  s t a t i s t i c s  o  Qijk '  f  ^ ( m l ) ' ^(m2+l)-'  s  (  i s  a  1 0 0 (  ' "  cc)% confidence interval of the slope estimator where m  1 =  (N-  Z . 1  m2 = (N+ Z _ x  (Var(S ))l/2)/ , ,  a / 2  2  m  (Var(S'))l/2)/2 ,  and Z j . / 2 is the (1 - a/2) quantile of the standard normal distribution. a  2.3 Median Polish Tukey (1977) proposed a procedure called median polish to fit a three-way model:  1 6  1  y.  = u + aj + bj + c + abjj + a  j k  k  C i k  + bc  jk  +e  i j k  (1)  where u is the common effect, a- is th ith row effect, 1  bj is the jth column effect, c  k  is the kth layer effect,  ab^ is the interaction of a- and bj, ac^ is the interaction of a- and c , k  bCj  k  k  is the interaction of bj and c  k  e^- is the random error of y^ . k  k  Roughly speaking, this procedure uses medians of the data to estimate the main effects and the interactions in the same way as means are used to estimate the effects in the case of A N O V A . Note that when we use means for fitting, the main effects and the interactions can be found in one "iteration", further iteration leaving the result unchanged. When we use medians, the first iteration may not be adequate.  Tukey (1977) developed one-way and two-way median polish as well while three-way polish is the generalization of two-way  procedure. Turkey pointed out that the  generalization of this three-way polish to a more-way case is straightforward. Here we describe the three-way median polish procedure only.  1 7  Suppose y . i s the observation under the ith level of factor a, the jth level of factor b, and the kth level of factor c(i = 1, —, I, j = 1,  J, k = 1, —, K). Let, in general, x  the median of any given set of numbers, X j ,  x ^ ; these may have additional subscripts  like x - j p x-j2>  x ^  k  where now x . j  M  amounts to computing the median over the last  M  subscript, and let  r  ijk yijk, =  f 0 r a l l i  'j' k  We fit the model (1) as follows:  (a) let 1 1 ^ = ^  =bj  =c  = ab^  k  = ac^ = bc  (b)forn = 0, 1 , 2 , l e t J3n+1) ij  a b  a  = a b  + r  (3n+l)_ (3n+0) i i a  . (3n+l) b  f3n+0) (3n+0) ij ij • M  _  j  u =  b  (3n+l)_  +  a c  (3n+0) j  (3n+0) iM (3n+0)  + b c  JM  (3n+0) (3n+0) +c  and  r  (3n+l)_ (3n+0) (3n+0) ijk " rjk " ijM r  denote  r  18  •  jk  = 0, r ^ = y  ijk  , foralli,j,k;  (3n+l) _ ik ~  a c  a c  (3n+0) ik ~  (3n+0) iM  a c  , (3n+l) , (3n+0) (3n+0) bc =bc -bc , jk  (3n+0)  (3n+2) ik  (3n+l) (3n+l) ik iMk '  = a c  (3n+2) i "  a  J M  (3n+l)_  a c  a  (3n+l) i  c  (3n 2)  u  +  = u  (3n+0)  + r  (3n+2) (3n+l) k ~ k  c  r  jk  (3n l) +  (3n+l) iM  + a b  + b c  + b  (3n+l) Mk  (3n l) +  5  (3n+2)_ (3n+,l) (3n+l) ijk ~ ijk " iMk r  .(3n+2) ij  a b  . b c  (3n+2) jk  = a b  = b c  r  (3n+l) ij -  (3n+l) jk "  a b  b C  ,(3n+l) iM •  (3n+l) Mk '  (3n+2) (3n+l) , ( 3 n + l ) bj =bj -b , M  1 9  (d) let . bc  (3n+3) ij  = b c  . (3n+2) (3n+2) ij MJk • +r  (3n+3) ,(3n+2) ,(3n+2) bj =bj ab , u  +  (3n+3) k  c  u  =  (3n+2) k  c  M J  + a c  (3n+2) Mk '  (3n+3) _ (3n+2) (3n+2) - u +a u  M  and  r  (3n+3)_ (3n+2) (3n+2) ijk ~ ijk " Mjk r  (3n+3) ij  r  u a b  a c  =  ab  ,(3n+2) ,(3n+2) ij " MJ a b  (3n+3) _ (3n+2) (3n+2) ik " ik " Mk a c  a c  (3n+3) _ ( 3 n + 2 ) (3n+2) a,i = - a^ o  (e) Repeat steps (b) to (d) until the sth stage, where s is the smallest number such that either (s) iJM  T  = a C  (s) iM  = b <  (s) JM  =  () M S  C  =  or  20  0  (s) iMk  or r.Mjk  Tukey (1977) suggested that two cycles of (b) - (d) would be enough — whether or not we are at the end of the process. He also suggested using the box plot to show the relative magnitude of the main effects, interactions and the residuals. Such a plot can display the significant main effects and interactions. Comparing median polish and mean polish (using means to estimate the effects), median polish minimize the sum of absolute values of the residuals while mean polish minimize the sum of squares of the residuals. Median polish is more robust than mean polish against outliers, namely, the median tends to leave outliers as spikes while the mean tends to reduce them and spread the deviations around. Whether or not the outliers are deleted from the data has a lesser effect on the result of median polish than on that of mean polish.  2.4  Kriging and Universal Kriging Kriging is the name given to best linear unbiased estimation of a stochastic process by  generalized least squares and its name comes from D. R. Krige, a mining engineer in South Africa. It is used to fit a surface by regression techniques.  21  Definition: A stochastic process Z(x) is said to be strictly stationary if the joint probability distribution function of n arbitrary points is invariant under translation (Delhomme, 1976). i.e.,  f(Z(xp, - , Z(x )) = f(Z(x h), n  1+  Z(x +h)) n  where h is arbitrary. Definition:  A stochastic process Z(x) is said to be weakly stationary if  its first two  moments are invariant under translation, namely, E(Z(x)) = constant  for all x  and Cov(Z(x)), Z(y)) = C(x - y), i.e., it depends only on x - y. Definition: The covariance of Z(x) is said to be isotropic if Cov(Z(x), Z(y)) = C(lx - yl), i.e., it depends only on the distance between x and y.  Definition:  A stochastic process Z(x) is said to be intrinsic if the first two moments of  Z(x+h) - Z(x) depend only on h. Definition: r(h) = j Var(Z(x+h) - Z(x)) is called semivariogram. 2r(h) is called variogram.  Observe that r(h) is essentially the negative of the covariance. If E(Z(x)) = constant,  22  then r(h) = j E[Z(x+h) - Z(x)]  2  and it may be estimated by (Delfiner, 1975)  where  = number of pairs of observation separated by h and Z(x^+h), Z(x-) are  observations. There are several commonly used variogram models when Z is istropic. One of those is called generalized covariance model which is used when E(Z(x)) * constant. Definition: m(x) = E(Z(x)) is called the drift of Z(x). Let e(x) = Z(x) - m(x); then Z(x) = m(x) + e(x),  (1)  where E(e(x)) = 0. In (1), e(x) represents the random fluctuation and m(x) represents the slowly varying, continuous features of Z(x). Therefore, m(x) may be approximated within a restricted neighborhood by L  (x) = X i=l  mi  a P  V  X )  (2)  23  where  are constant coefficients and  are arbitrary functions of the point x. In particular,  if the (f-(x)} are spatial polynomials, Kriging is called "universal Kriging".  From (1), it can be seen that to estimate m(x) from the data we need to know the covariance of e(x), and to estimate e(x) we need to know m(x). But usually in practice, neither is known and both must be estimated from the data. Matheron (1973) and Delfiner (1975) have developed a technique to estimate m(x) and the covariance of e(x) simultaneously; the key concept is the generalized increment. Definition: A generalized increment of order k is a linear combination of the sample value 6-Z(x.) for which  y, B-f(x-) = 0, where f(x.) is a polynomial of order less than or equal  to k. In the plane (x. = (x,., x~.)), this condition yields:  k = 0,  k=l,  X i XVli=XV2i = ' 6  i  k = 2,  0  =  i  i  X i X Vii = X i 2i = X %4 = X i 2i = X ¥li 2i = °-  i  B=  i  i  6 X  i  i  B x  i  x  A  _  A n increment of order k will filter out a polynomial trend of degree k. If Z = 2_, X | Z(x-) i A  is a Kriging estimate of Z(x), then Z(x) - Z , the Kriging error, is a generalized increment since it can be written as -Z(x)  and we see that BQ = -1 and B- = 7^ for all i.  ^ Z(xp i  Definition: If there exists a function K(h) such that for any kth order generalized increment  X  B  i  Z ( x  i ' }  i  Var( £  BJZCXJ) ) = £  ^ i j ( i " j) 6  K  x  ( )  x  3  then K(h) is called a generalized covariance function, and Z(x) is known as an intrinsic process of order k.  Matheron (1973) shows that any isotropic(h = Ihl) generalized covariance function defines a class of functions which are all equivalent up to an addition of an arbitrary evenpowered polynomial of degree less or equal to 2k.  For example, for k = l , K(h) = -h and  2 K(h) = -h + h  are equivalent. F o r this reason, only the essential odd powers are used  when writing a generalized covariance.  For orders up to 2, the isotropic polynomial  generalized covariance kernels are (Jernigan,1986): C5 + a h,  k= 0  Q  K(h) =<  3  CS + aph + a j h C8 + a h + a j h  , 3  Q  where ag, a^, a  2  ^ 0, and in R  X 5  ~ '10,  k=l  + a h , 5  2  (4)  k = 2  , a^ > - 1 0 ^ a n a /3, in R , a^ > -^j 10aga , 2  h= 0 h* 0•  25  2  The constraints insure that (3) is nonnegative. Polynomial generalized covariances with coefficients satisfying these constraints are said to be admissible.  Our objective is to estimate Z(x) by Z = ] ^ X]Z(x^) such that  J E(Z(x) - Z(x)) = 0  (5)  I Var(Z(x) - Z(x)) = E(Z(x) - Z(x)) is minimized. 2  Consider the case of planar region(x = (x^^)).  Using the previous assumption of a  polynomial drift of order less than or equal to k,  m(x) =  a f (x), p  p  2  2  where f (x) are given by { 1 } for k = 0, { 1 , x^, X 2 } for k = 1, { l ^ ^ ^ , X £ , x-^} p  for k = 2,  it has been shown that the X's satisfy (5) can be found by solving the following system of equations, known as the universal Kriging system:  <  X  *iK(x. - j) + x  X  Vp i ( x  } =  K ( X  i"  X )  f O T a  U  1  (6)  Y, X:f ~  1  (x.) = f (:x) P J P  for all p.  If we know (a) the order of the drift k, and (b) the coefficients of the generalized covariance function, we can solve (6) to estimate (Kriging) Z(x). (6) has a unique solution for X[, provided that the drift functions f are algebraically linearly independent, i.e.,  a f (x) = 0 p p  for all x if and only if a = 0 for all p. P  26  (a) Drift order identification Here is a procedure to determine the order of drift k (Devary and Rice, 1982): for different values of k, assuming Z ( x ) is an intrinsic function of order k, delete each (k) observation Z ( x ) in turn and calculate Z  ( x p , the estimate of Z ( x ) , from the remaining  f  r  observations using the generalized covariance function K(h) = - h. This particular generalized covariance function is valid for any value of k. Typically k is chosen to be 0, 1, or 2. (k) The best choice of k depends on the residuals ( Z ( x - ) - Z  ( x p , all i}, k = 0, 1, 2. Three  criteria are set up to compare the residuals: (k) 1. rank {IZ(x.) - Z  (x.)l, all i, k = 0, 1,2}. The value of k with the smallest average rank  is the preferred order of drift value; 2.  the value of k with the smallest M S E is preferred, where  MSE  k  =X i  (Z(xp " Z  ( k )  (  X i  )),  3. the value of k with IZjJ < 1.96 is preferred, where  e= Z(Z(x )-Z i  ( k )  (x )), i  27  k = 0, 1, 2;  1  S(e ) = [X  (ZCxp - Z  ( k )  (  X i  ) -  ) /(n - l ) ] , 2  e  2  since under the hypothesis that the true value of k is less than or equal to k, Z ^ is asymptotically N(0, 1). So, typically, a k with minimum M S E and/or average rank is selected.  (b) Selection of the generalized covariance function. For a chosen k, the model for a polynomial generalized covariance function of order k is k K(h) = C 6 + X  a lhl  2p  h=0  where  h*0 •  p=0  Delete the rth observation from data, then estimate Z(x ) with the initial estimates of the r  coefficients {C = 1, a = -1, a, = 1, a~ = -1}. For example, if k = 1, the initial estimates of n  K(h) is K(h)  1  -h+h . 3  Suppose  J  is the error of the Kriging estimate of Z(x ); by equation(3),  28  Var(Z (8)) = E(Z (B)) = £  B ^ B ^ K ^ - )  2  r  r  (6)  Xj  ij  or  E(Z (B)) . c £  ( B f >)  2  ta X  +  2  r  j  p  P=l  %  ( l )  Bf  \ -  x,! ** 2  1  iJ  (7) Let  j  ij  Equation (7) can be written as  k  E(Z (6)) = 2  r  CT +Xa T p  0  p  (8)  p=l  which is linear in the coefficients. To determine the coefficients, Devary and Rice (1982) 2 r r suggest regressing E(Z (B)) r  upon T Q ,  T^with the corresponding constraints on the  coefficients shown in (4). The set of coefficients obtained by this regression is used to reestimate the observations, yielding a new set of errors and T* values. The regression,  29  (8), is performed again and again until the parameters converge. The proposed iteration is to be performed on all of the generalized covariance function of order k.  Once several sets of admissible parameters have been estimated it remains to choose 2 the best one. Since Z (6) is a Kriging error and E(Z (B)) is a estimate of the Kriging 2 r  r  variance, if the estimate of K(h) is correct, E(Z (B)) r  should be close to Var(Z (B)) r  defined by (6). Therefore, the quantity  P=I( ( )) /I°r Z  B  2  r  r  r  should be close to 1, where o'  B- B j K(x. - x.). In practice, a jackknife estimator of T  1  l  p is recommended by (Delfiner,1975): p = 2R - (n r + n~r ) / (n, + n ), 1  1  9  9  where  R =  Xz (B) /X"?. 2  r  r  r  5X<B) /]T<£, 2  reL  re I  r = £z (B) / £ Q 2  2  2  R  ,  n, = number of r's such that r e I 1,  30  = number of r's such that r e L^. It is sufficient to let I^ = {1, —, n/2},  = {n/2 + 1, —, n} where n is the total number of  observations. The generalized covariance is chosen as the one with the minimum M S E and jacknife statistic p e (0.75, 1.25). For n < 50, the minimal M S E criterion should be used.  There have been some objections and reluctance to use this technique because the resulting covariance function is difficult to interpret. Journel and Huijbregts (1978), Devary and Doctor (1981), as well as Neuman and Jaeobson (1984) have suggested different approaches to estimate the covariance function.  31  Chapter 3  APPLICATIONS AND O V E R V I E W  Identical statistical analyses are applied to the "historical" data collected from 1980 to 1986 and the "recent" data collected from 1983 to 1986 for three ions: S 0 , NOg and H  +  4  In  general, "recent data" is just a part of the "historical data" except that more than doubled number of sites are involved in the former.  3.1 Transformation and Clustering A preliminary exploration of the distribution of the data provides important information which enables us to analyze the data efficiently and accurately. And transforming the data so that they are approximately normally distributed also makes the analysis simpler. Three commonly used transformations, y=x / , y=x^ and y=log(x) are examined separately. 1  2  4  It  seems that in most of the cases, the log transformation is the best, in the sense that the resulting empirical distribution is approximately symmetric.  However the empirical  distribution does have relatively long tails compared with the normal. This is similar to previous conclusions about transforming environmental data (cf. Gilbert, 1987, pl52). The hierarchical clustering analysis with complete linkage described in Section 2.1 can be applied to either the untransformed or transformed data to cluster the sites under consideration.  The difference is that if we cluster the log transformed data we can expect  more clusters among the sites with smaller measurements measurements.  than those with larger  This is because log is a concave function. Since, in general, we are more  interested in the sites with larger measurements,  and we want to study them more  carefully, we decided to cluster the untransformed data.  32  Furthermore, since the  precipitation volume weighted mean gives the quantity of chemicals in wet deposition rather directly, we apply this method to the volume weighted mean for all the components under study. For the three ions under study, the results of clustering analysis show that all the sites can be grouped into about three clusters (though  the N O ^ data from 1980 to 1986 is  actually grouped into two clusters), namely, the subregion of the United States with the highest concentration of industry, a roughly concentric area around that region, and the rest of the United States. This result agrees with common sense; industries which emit more SO~ , NO  and thus have more concentrated  emission patterns should be more or less  contiguous.  After clustering the data, the histogram of the log transformed data is drawn for each cluster. It turns out that in most of the cases, the data within clusters have histograms are approximately symmetric but with relatively long tails.  Sometimes these tails are heavy  and sometimes not. This suggusts some uncertainty about whether or not the data can be treated as normal. However, our analysis does not rely on normality so this issue is not of great concern. The remainder of our analyses are based on log transformed data, for both monthly precipitation volume weighted means and the monthly medians though only the histograms of the former are examined.  We refer the log transformed data simply as "data" in the  remaining parts of this chapter.  3.2  T r e n d a n d Seasonality  Since the normality of the data is doubted, as many cases when people are dealing with precipitation chemistry data, using the parametric methods such as analysis of variance to  33  estimate trend and seasonality may not be appropriate.  But  this does not cause any  difficulties in using polish method. In addition, the median is resistant to outliers, which appear in our case quite often. Therefore we decided to use median polish to extract out the trend and the seasonality of the data. The three-way median polish method described in Section 2.2 is applied to both the "historical" and "recent" data, as well as to both the monthly volume weighted mean and monthly median for all three ions. For each ion, either "historical data" or "recent data", and either monthly volume weighted mean or monthly median, model (2.3.1) can be rewritten as C = u + Mj + Y .  +  S  k +  M Y j . + MSfc + Y S  j k  +  e  i j k  where u is the common effect, M- is the ith monthly effect, i = 1,  ,  (3.2.1) 12 standing for Jan.,  Feb., —, Dec. respectively, Yj is the jth yearly effect, where for the "historical" data set, j = 1, —, 7 stands for 1980,1981, —, 1986 respectively, and for the "recent" data, j = 1, —, 4 stands for 1983, 1984,  1986, respectively, S is the kth site effect, k = 1,  with "historical" data, K = 86 for H 32 for H  +  k  +  K , where  and K = 81 for the other ions. With "recent" data, K =  and K = 31 for the other ions, and e-j is the residual. k  The estimated main effects, namely, monthly, yearly and site effects are plotted. Residuals are displayed using boxplots where each box corresponds to a site. A summary of the three-way polish is given using boxplots where each box represents a source of main effects, interactions or residuals. This summary enables us to see how big the main effects and interactions are compared with the residuals.  3.3  N o n p a r a m e t r i c Test, Slope Estimate a n d K r i g i n g  34  Some characteristics of precipitation chemistry data, such as nonnormality, the existence of missing data and the limited number of observations along the time, create some difficulties in using the traditional parametric statistical methods to test the trend in such cases. But these cause no difficulties in using the nonparametric trend tests described in Section 2.2. This is why nonparametric trend test is used in this study. As mentioned in Section 2.2, for data with seasonal patterns, the test procedure which deseasonalizes the data first and then ranks them altogether can preserve the relative ranks between seasons; but these are lost if the data are ranked within each season. Consequently, Farrell's (1980) test is more powerful.  Considering that the length of the  "recent" data record is merely 4 years, to test the trend over time we prefer Farrell's (1980) test. However, since there are some missing values in the data set which would make the computations very complicated, we decided instead to deseasonalize the data first and then use the Mann-Kendall Test to test whether or not there is a monotone trend over time in the deseasonalized data.  For each individual site, the data are deseasonalized using the  one-way median polish by subtracting the median of  the values of each month from the  data for that month. Then the Mann-Kendall Test is applied to the deseasonalized data to obtain the test statistics and the corresponding p-values. Sen's nonparametric slope estimates are obtained with 80% confidence intervals. Since the alternative hypothesis of the test is that the trend is monotone, for some of these tests our failure to reject the null hypothesis may result from the V-shaped trend in the data. This is suggested in particular by the plots of the yearly effects of some data obtained from the three-way median polish (for example, the monthly volume weighted mean of SO4 for 1980 - 1986).  Similarly, a  small slope estimate could be derived from the V-shaped trend.  Nonparametric slope estimates are plotted on the map of the United States. If the lower 80% confidence bound of the slope estimate at a point is greater than 0 then a "+" is plotted  35  at that point. If the upper 80% confidence bound of the slope estimate at a point is less than 0 then a "-" is plotted at that point. . Otherwise a "0" is plotted on the point. If the p-value of the Mann-Kendall Test is less than 0.2 at a point, then an "x" is plotted on the map of the United States. The resulting plots are quite similar to the plots produced by the slope estimates of the trend if we convert "+" and "-" in the latter into an "x". This is not surprising since we expect that the 80% confidence interval of the slope estimate and the p-value of the Mann-Kendall Test at the a = 0.2 level to give us about the same information. The plots of p-values for the Mann-Kendall Test are not included in the report since the plots produced by the slope estimates of the trend contain more detailed information. Based on the slope estimates of the trend, universal Kriging estimates of the trend with estimated standard errors are obtained at each integer degree grid point of longitude and latitude across the 48 continental states in the United States.  The estimates and the  estimated standard errors are calculated for the nearest 8 neighbors of the point being estimated for the "historical" data and for the nearest 10 neighbors of the point being estimated for the "recent" data.  Note that usually the estimates and the standard errors  estimated by Kriging are based on actual observations rather than some kind of estimates treated as data.  So we have to interpret them with caution since the estimates and the  estimated standard errors are different from the real Kriging estimates and the standard errors.  The results of Kriging are plotted on the map of the United States. If the upper one standard error bound is less than 0 at some point then a "-" is plotted at the point. If the lower one standard error bound is greater than 0 at some point then a "+" is plotted at the point. Otherwise nothing is plotted.  36  Chapter 4  ANALYSES ANDCONCLUSIONS  FOR SULPHATE  Monthly precipitation volume weighted mean and monthly median are are referred to as "weighted mean" and "median", respectively, below.  4.1  Results for the Historical D a t a  "Historical data" refers to the data collected from January, 1980 to December, 1986. For S O ^ , there are 31 sites with 5 or fewer missing observations during the period. locations of these 31 sites are plotted in Figure 4.1.1.  The  From this figure we can see that  more sites are located in the East than the West. The results of this section are based on the data obtained from these 31 sites.  4.1.1  Clustering a n d T r a n s f o r m a t i o n  Before clustering the data, one outlier was deleted from the data set, namely the observation at the site with ID=040a (see Olsen and Slavich, 1986), observed in September, 1986.  Its value is 71.71, much larger than 3.2, the average value for that site.  Figure 4.1.2 (a) shows the hierarchical cluster structure of the weighted means. The sites are partitioned quite naturally into three clusters. cluster are shown in Figure 4.1.3.  The locations of the sites labelled by  The pattern is obvious: cluster 2 is located in the  subregion of highest industrial concentration; cluster 1 is located around that region; and cluster 3 is spread over the rest of the United States.  This indicates that the industrial  areas emit m o r e S 0 2 than other areas and the pattern of emission is consistent with intuition. A cluster analysis was done without deleting outliers. The clusters are the same however, except for site 040a which is ruled out of any cluster (Figure 4.1.2 (b) ).  37  The histograms of the original data and the transformed data under the three kinds of transformations described in Section 3.1, are shown in Figure 4.1.4.  It appears that the  histogram of the log transformed data fits the normal curve best. The other three are right skewed. So the log transformation is adopted in this case and the remainder of our analysis of  the sulphate data is based on the transformed data.  The histograms of the log  transformed data for each cluster are depicted in Figure 4.1.5. The extremely long right tail in the histogram of cluster 2 results from the outlier at site 040a.  In general, these three  histograms are not unduly skewed but do have relatively long tails.  4.1.2  Trend, Seasonality and Spatial Patterns  The analysis described below is done for both log transformed monthly volume weighted mean and log transformed monthly median SO4 data. The analysis is applied separately to each cluster obtained from clustering the volume weighted mean data. Three-way median polishes are applied to both weighted means and medians in each cluster, in the manner described in Section 3.2.  Figures 4.1.6 and 4.1.7 show the plots of  the yearly effects of the three clusters for weighted mean and median, respectively. They show similar patterns although the one for weighted means is more obvious.  A l l of the  three clusters show down-trends before 1983 and cluster 3 shows a down-trend after 1983 as well.  But clusters 1 and 2 show up-trends after 1983.  This indicates that the  concentration of SO4 in wet deposition increased after 1983 for the most industrialized subregion of the United States and that the down-trends for the rest of the United States have not changed much up to 1986. The yearly effects of the medians are more pronounced than those of the means in 1980 and gradually became smaller than the former in 1986. This indicates that in 1980, more than half of the sites had yearly effects larger than the average of their corresponding cluster, while in 1986, the situation was reversed: less than half the sites had the yearly effects larger than the average of the corresponding cluster.  38  Figure 4.1.8 and Figure 4.1.9 show the monthly effects for the weighted means and medians, respectively.  They have almost the same pattern .  The patterns for all three  clusters are similar as well. The effects are high in summer and low in winter; spring and autumn fall in between and the transitions are quite smooth.  The fact that the seasonal  patterns are very similar for all three clusters suggests a reassuring spatial stability in this seasonal pattern. But the complexity of the acid deposition process makes it unreasonable to infer that the emission of S C ^ has the same pattern. Figures 4.1.10 to 4.1.12 display the site effects of log(SO^) by cluster; the weighted means and medians are plotted together and sorted by weighted means within each cluster. The site effect of Station 039a for weighted means in Figure 4.1.10 are quite different from that of the medians. This may be caused by the fact that concentration tends to be smaller when the volume of the precipitation is greater. For example, if c^ = C 2 =10 and c^=l are the concentrations, while v^ = V 2 = 1, and v^ = 10 are the volumes, the median of c l , c2 and c3 is 10 but the volume weighted mean is about 1.36.  Figures 4.1.13 to 4.1.15 are the boxplots of the residuals of the three-way median polish of the log(SO^) data where (a) is for weighted means and (b) is for medians and each box represents one site.  In general, the variations of the residuals in each cluster are of the  same order and comparable. The differences between the boxplots for the means and for the medians are even smaller. In Figure 4.1.13 ((a) and (b)), we do observe a few extreme outliers from Station 039a, and from Station 049a and Station 168a as well. For these sites, further detailed study is needed. In Figure 4.1.14 ((a), (b)), an extreme outlier is observed in the box for Station 040a. This is the one which was deleted when we did the clustering. It was not deleted here because it cannot affect the result of the median polish by its extremely large value.  In Figure 4.1.15 ((a), (b)), the boxes for Stations 059a and 074a  39  have extreme outliers and they need a more detailed study as well.  But no analyses for  specific site are included in this paper. Figures 4.1.16 to 4.1.18 summarize the median polish for the three clusters, respectively where (a) is for the weighted meana and (b) is for the medians.  In each figure, boxes  represent the main effects, interaction and residuals respectively.  We regard interactions  as non-negligible if the sizes of the boxes for the interactions are comparable with those for main effects.  In Figures 4.1.16 (a) and (b), the boxes for monthly effects are relatively  large indicating that the variation caused by seasonality is larger than the other effects. The boxes for the interactions of yearly effects and site effects are relatively small indicating that the trends for all the site in the cluster 1 are similar. This convinces us that the pattern of trends is relatively stable. The boxes for the other interactions are similar in size to those of the main effects, which suggests that they have to be taken into account. For cluster 2, the same thing happens again, namely the monthly effects are relatively large and the interactions between yearly effects and site effects are relatively small (see Figures 4.1.17 (a) and (b)).  Figures 4.1.18 (a) and (b) show the summaries for cluster 3.  We observe relatively large monthly effects and relatively small year and site interactions as before. The interpretation for cluster 1 applies here as well.  4.1.3  T h e Results of T r e n d Testing, Slope Estimation a n d K r i g i n g  Mann-Kendall Test and Sen's nonparametric slope estimation procedure are applied to the deseasonalized data for each site in the manner described in Section 3.3. Kendall  A Mann-  Test statistic with the corresponding p-value and Sen's (1968b) nonparametric  slope estimate with its 80% confidence interval are obtained for each site for both the weighted means and medians. These are shown in Tables 4.1.1 and 4.1.2 respectively.  In  Table 4.1.1, 24 out of the 31 sites have significant trends at the p=0.2 level. In Table 4.1.2, 22 out of the 31 sites have significant trends at the p=0.2 level.  40  The results of the nonparametric slope estimation are plotted in Figures 4.1.19 (a) and (b) with symbols defined in Section 3.3, for the weighted means and medians respectively. If the lower 80% confidence bound of the slope estimate is greater at any site than 0 then a "+" is plotted; if the upper 80% confidence bound of the slope estimate is less than 0 a "-" is plotted. Otherwise a "0" is plotted . "+", "-" and "0" represent respectively, an up-trend, a down-trend and no-trend. The patterns of these two figures are very similar. About 2/3 of the sites show down-trends and they are mainly located in the Eastern United States. The other 1/3 of the sites show no-trend and no site had an up trend. During the period of 1980 - 1986 most of the sites have down trends.  This conclusion agrees with some  previous findings that the wet deposition concentration of S O ^ had decreased from late 1970's to early 1980's (Schertz and Hirsh,1985, Dana and Easter, 1987, Seilkop and Finkelstein, 1987).  However, as we mentioned before, this could be a misleading  conclusion in the presences of a V-shaped trend. At the very least, this result says that the concentrations of SO^ for the latter years of the sampling period are smaller than that of early years of the same period. The results of Kriging the slope estimates are displayed in Figures 4.1.20(a) and (b) As before, Figure 4.1.20(a) is for the weighted means and (b) is for medians. A "+" means an up-trend is estimated at the corresponding point and a "-" means a down-trend, estimated at the point. Again, these two plots are similar. Both Figures 4.1.20 (a) and (b) show that there is a down-trend for the most of the United States except that a few small areas in the Eastern United States show no trend.  Considering that these results were  obtained from the same set of data which produce Figures 4.1.19 (a) and (b), this is quite natural.  The difference between these two figures are probably caused by (a) the  differences in magnitudes of the slope estimates of weighted means and of medians, or (b) the fact that the estimated order of the variograms is one for the weighted means and zero for the medians. Note that these two figures can only be used as a guideline for the spatial  41  distribution of the trends and should be read in conjunction with Figure 4.1.19.  At any  particular point where the observations were not made, interpolation may not be reliable. This is because (a) estimation is based on the estimated slopes, not on the observations directly and (b) the spatial resolution of the measurements is low and therefore, the correlations between sites are small.  4.2  Results for the Recent Data "Recent data" refers to the data collected from Jan., 1983 to D e c , 1986. For SO^, there  are 81 sites with 5 or fewer missing observations during the period. The location of these 81 sites are plotted in Figure 4.2.1. From this figure we can see that more sites are located in the Northeast than the rest of the areas.  The results of this section are based on the  data observed at these 81 sites.  4.2.1 Clustering and Transformation Before clustering the weighted mean data, two outliers were deleted from the data set, namely the observations for the Stations 035a and 040a (see Olsen and Slavich, 1986). The first one is observed on January, 1985 with value of 365.56, much larger than 6.39 which is the second largest observation at the same site; the second one is observed on September, 1986, the same one mentioned in Section 4.1.1.  Figure 4.2.2 (a) shows the  hierarchical cluster structure of the weighted means. The sites partition naturally into three clusters except for Stations 070a and 071a. These two sites are assigned to cluster 3, the closest one according Figure 4.2.2 (a), for technical convenience. The locations of the sites labelled by cluster are shown in Figure 4.2.3. It gives a pattern similar to the one in Figure 4.1.3: cluster 1 is located in the subregion of highest industrial concentration; cluster 2 is located around that region; and cluster 3 is located throughout the rest of the United States,  42  corresponding to cluster 2, 1 and 3 in Figure 4.3.1 respectively.  This gives us the same  indications we mentioned in Section 4.1.1, namely, that the industrial areas emit more S C ^ than other areas and the pattern of emission is consistent. without  A cluster analysis was done  deleting outliers. The clusters are the same except for Stations 035a and 040a  which are ruled out of any cluster (Figure 4.2.2 (b)).  The histograms of the original data and the transformed data under the three kinds of transformation described in Section 3.1, are shown in Figure 4.2.4.  It appears that both  histograms of the log and 1/4 power transformed data fit the normal curve. The other two are right skewed.  We adopt the log transformation in this case since it is the best  transformation in a lot of other cases, particularly in the case of "historical data". The remaining analyses of this section are based on the log transformed data. The histograms of the log transformed data for each cluster are depicted in Figure 4.2.5.  Again the  extremely long right tail in the histogram of cluster 1 results from the outlier at site 040a (the outlier at site 035a is not included in the plots).  The histograms for the other two  clusters have long left tails.  4.2.2  T r e n d , Seasonality a n d Spatial Patterns  The analysis described below is applied to both log transformed monthly volume weighted means and log transformed monthly medians. The analysis is applied separately to each cluster obtained from clustering the volume weighted mean data. Three-way median polishes are applied to both weighted means and medians in each cluster. Figures 4.2.6 and 4.2.7 show the plots of the yearly effects of the three clusters for weighted mean and median respectively, which show similar patterns.  Clusters 1 and 2  show up-trends while the trend of cluster 3 is probably going down.  This conclusion,  obtained from 81 stations instead of 31, is consistent with what we got in Section 4.1.2.  43  That is that the concentration of S O 4 in wet deposition increased  for the most  industrialized subregion of the United States during the period of 1983 to 1986 though the precise change point in 1983 is not clear yet. In contrast to the situation in Figure 4.1.6 and 4.1.8, it is hard to say the yearly effects of the weighted means or those of the medians is larger.  Figures 4.2.8 and 4.2.9 show the monthly effects for the weighted means and medians, respectively.  Although more than half of the data used in the present analyses were not  included in the historical data, the patterns are almost the same as those for the latter displayed in Figures 4.1.8 and 4.1.9: high in summer, low in winter with spring and autumn in between. This convinces us again of spatial stability in the seasonal pattern. Figures 4.2.10 to 4.2.12 show the site effects of log(S0 ) by cluster; the weighted 4  means and medians are plotted together and sorted by weighted means within each cluster. For cluster 1, the stations located in the center of the industrial region tend to have larger effects. No unusual results are found in these figures. Figures 4.2.13 to 4.2.15 are the boxplots of the residuals of the three-way median polish of the logCSO^) data where (a) is for weighted means and (b) is for medians and each box represents one site.  In general, the variations of the residuals in each cluster are of the  same order and comparable, and comparable to those in Figures 4.1.13 to 4.1.15 as well. The differences between the boxplots for the means and for the medians are even smaller. Extreme outliers are found in Stations 040a, 168a, 249a and 350a in Figures 4.2.13 (a) or (b), 039a and 049a in Figures 4.2.14 (a) or (b) and 035a, 059a, 074a, 255a, 271a and 354a in Figures 4.2.15 (a) or (b). Among these stations, 039a, 049a, 168a, 040a, 059a and 074a have extreme outliers observed in Figures 4.1.13 to 4.1.15 and those outliers may be the same ones. A l l these sites need further study.  44  Figures 4.2.16 to 4.2.18 summarize the median polish for the three clusters respectively where (a) is for weighted means and (b) is for medians. In each figure, boxes represent the main effects, interaction and residuals. The main difference between these figures and Figures 4.1.16 to 4.1.18 is that the variation of the yearly effects for 1983 to 1986 is much smaller than those for 1980 to 1986. In Figures 4.2.16 (a) and (b), the monthly effects are larger than the other effects as in Figures 4.1.16 ((a) and (b)) while the yearly effects and the interactions of yearly effects and site effects are relatively small. Cluster 2 has the same characteristics as cluster 1 (see Figures 4.2.17 (a) and (b)). (b) show the summaries for cluster 3.  Figures 4.2.18 (a) and  That the station effects are relatively large is  perhaps due to assigning Stations 070a and 071a to this cluster. In general, the trends for 1983 to 1986 are small in magnitude and not very consistent for all sites.  4.2.3  The Results of Trend Testing, Slope Estimation and Kriging  The Mann-Kendall Test and Sen's slope estimation procedure which were applied to "historical data" are used for "recent data" as well. A Mann-Kendall test statistic with the corresponding p-value and Sen's (1968b) nonparametric slope estimate with its 80% confidence interval were obtained for each site for both the weighted means and medians. These are shown in Tables 4.2.1 and 4.2.2 respectively.  In Table 4.2.1, 40 out of the 81  sites have significant trends at the p=0.2 level. In Table 4.2.2, 31 out of the 81 sites have significant trends at the p=0.2 level. Figures 4.2.19 (a) and (b) are obtained by the same method used to produce Figures 4.1.19, for weighted means and medians respectively. Again, "+", "-" and "0" represent an up-trend, a down-trend and no-trend, respectively.  In Figure 4.2.19 (a), about 1/4 of the  sites show up-trends and are mainly located in the Eastern United States. Approximately 1/4 of the sites show down-trend and are located mainly in the west and middle of the U . S. The rest of the sites show no-trend and are located throughout the United States.  45  The  pattern of Figure 4.2.19 (b) is basically the same except that there are more "no-trend" stations and fewer "up-trend" stations.  This probably is due to the fact that usually the  estimates based on medians is more conservative than those based on means.  The  patterns in these two figures are consistent with the results of median polish since the site with "-"'s belong mainly to cluster 3 and those with "+"'s, to clusters 1 and 2. Comparing Figures 4.2.19 ((a) and (b)) with Figures 4.1.19 ((a) and (b)), the conclusion is that though the trends for the stations located in the Western United States continue to decline, most of the stations located in the Eastern United States no longer have down-trend and some of them have an up-trend during 1983 to 1986! This implies that some of the stations do have a V-shaped trend. The results of Kriging the slope estimates are displayed in Figures 4.2.20(a) and (b) where (a) is for the weighted means and (b) is for medians. As before, a "+" means an uptrend and a "-" means a down-trend estimated at the corresponding point, respectively. Again, these two plots are similar except that both the "+" area and "-" areas in (b) are smaller than those in (a). Both Figures 4.2.20 (a) and (b) show that there is a down-trend for the most of the Western United States, an up-trend in certain area of the Eastern United States and no-trend in the rest of the areas.  Comparing these two figures with Figures  4.1.20 ((a) and (b)), we can see that the areas with "-" shrink significantly towards the West and the areas with "+" appear in the East.  The difference in the patterns of the  trends is quite remarkable. For the reason given in Section 4.1.3, these two figures can only be used as indicators of the spatial distribution of the trends as the interpolation error for a specific point could be big.  46  The Locations Of The 31 Monitoring Stations From 1980 To 1986 (For  Sulphate)  Figure 4 . 1 . 1  Clustering of S 0 4 monthly volume weighted mean based on sqrt(MSE)  1980- 1986  4*  CO  §  §  S 9 6  Figure 4.1.2(a)  1  I  1  I  rt  CO  8  t  8  8  Clustering of S 0 4 monthly volume weighted mean based on sqrt(MSE) 1980 - 1986 (with  outliers)  CD  8  fl  $ 8  n  8  3 S  Figure 4.1.2(b)  8  3  3  »  W  OJ  O  8  8  Clusters of S 0 4 monthly volume weighted mean based on sqrt(MSE) 1980- 1986 (k=3)  Figure 4.1.3  Histograms of Transformed S 0 4 (Volume Weighted M e a n , 80-86) Histgram of S04 (volume weighted mean, 80 - 86)  Histgram of log(S04) (volume weighted mean, 80 - 86)  8 8  Histgram of sqrt(S04) (volume weighted mean, 80 - 86)  -I  Histgram of (S04) (1/4) (volume weighted mean, 80 - 86) A  S •  —i 0.5  Figure 4.1.4  1  1  1  1  1.0  1.5  2.0  2.5  1— 3.0  Histograms of log(S04) by Clusters (Volume Weighted Mean, 80-86) Histogram for Cluster 1 of log(S04) (vwm, 80 - 86) based on S04  8  8 -  ro  Histogram for Cluster 3 of log(S04) (vwm, 80 - 86) based on S 0 4  2  3  Figure 4.1.5  Histogram for Cluster 2 of log(S04) (vwm, 80 - 86) based on S04 n  Yearly Effect of log(S04) for 3 Clusters (monthly volume weighted mean, 1980 - 1986)  m  o  I  1980  —  1981  1982  1983 Figure 4.1.6  53  1984  1985  1986  54  Monthly Effect of log(S04) for 3 Clusters (monthly volume weighted mean-, 1980 - 1986)  1  Jan  1  Mar  1  May  1  . Jul  Figure 4.1.8  55  1  Sep  1  2  Nov  Jan  Monthly Effect of log(S04) for 3 Clusters (monthly median, 1980 - 1986)  2  1  Jan  1  Mar  1  May  1  Jul Figure 4.1.9  56  1 Sep  1  Nov  2 Jan  Station Effect of log(S04) for Cluster 1 (monthly median/volume weighted mean (sorted by volume weighted mean), 1980 - 1986)  CM d  Common Effect for Mean Common Effect for Median  71  0.728 0.834  A\ 7 / / /  / /  \  cn  /  I  /  c o  /  a  / /  \ \ \ \ \  7f\ /\ /\ /\ /\ /\  7 /  \  /  \  /  \  /  /  \  /  /  CM  9  /  \  /  /  c>  <8 CM IO  o  median  /  mgan  / /  \  CO  LA  / \ \ / \ / \ / \ / \ / \  /  \  \ \  N  /  /  / ID  o  o  stations  Figure 4.1.10  CO in  CO CM  Station Effect of log(S04) for Cluster 2 (monthly median/volume weighted mean (sorted by volume weighted mean), 1980 -1986) Common Effect for Mean = Common Effect for Median =  1.094 1.212  v V  \ \  /  / \ /\ V\  / / / / / / /  A o  V  \ \ \ \  / \  7  \ \ \  A  / /  median  03  to to  ro in o  stations Figure 4.1.11  mian  <8 to in o  Station Effect of log(S04) for Cluster 3 (monthly median/volume weighted mean (sorted by volume weighted mean), 1980 - 1986) Common Effect for Mean = Common Effect for Median =  0.231 0.220 V  \ \ \ / \  >  V  J  / \  \  Ar*  EL  r~i  A  \  /  m*dian  mian  / \ / / re  re cn io o  re CO  (0  5  re o  CO  o  re re  MCO  o  stations  Figure 4.1.12  CO  o  re  o o  re  CO CO  Boxplot for the resid. of log(S04) (80-86, monthly volume weighted mean, clust 1)  station Figure 4.1.13(a)  Boxplot for the resid. of log(S04) (80-86, monthly median, clust 1)  3  ro  cn  cn  CO station  Figure 4.1.13(b)  ra eg m  Boxplot for the resid. of log(S04) (80-86, monthly volume weighted mean, clust 2)  station  Figure 4.1.14(a)  Boxplot for the resid. of log(S04) (80-86, monthly median, clust 2)  station Figure 4.1.14(b)  Boxplot for the resid. of log(S04) (80-86, monthly volume weighted mean, clust 3)  station  Figure 4.1.15(a)  Boxplot for the resid. of Iog(S04) (80-86, monthly median, clust 3)  cn  o w  t I  I  1  I I  1 !  I I  i  ra o  1  CO  3  1  CO  o  ra  CD CO O  station  Figure 4.1.15(b)  ra  CD O  CO CO te-  Summary of the Effects and Residuals from Median Polish of log(S04) (80-86,monthly volume weighted mean,clust 1)  Sit  MonYr Figure 4.1.16(a)  MonSit  Summary of the Effects and Residuals from Median Polish of log(S04) (80-86,monthly median,clust 1)  Sit  MonYr Figure 4.1.16(b)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(S04) (80-86,monthly volume weighted mean,clust 2) •  • •  -  •  i  • 1  1 1  ~  x  1 1  | 1  1  =  1 1  T I— —| i 1  i  i  I  J_  1  • T  •  i L  1 —r— 1  }  •  i  1  I •  .  ii  i i  T i •  Mon  Yr  Sit  MonYr Figure 4.1.17(a)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(S04) (80-86,monthly median,clust 2)  t  Mon  Yr  Sit  MonYr Figure 4.1.17(b)  MonSit  t i  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(S04) (80-86,monthly volume weighted mean,clust 3)  i T  T  ~~T~  1  <=F=  i  Mon  Yr  Sit  MonYr Figure 4.1.18(a)  1  1  MonSit  i  1 1  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(S04) (80-86,monthly median,clust 3)  i !  i  I 1  X  i  i 1  1  i  1  1  i  I  I •  Mon  .  Yr  Sit  MonYr Figure 4.1.18(b)  MonSit  YrSit  res  Trend of log(S04) from 1980 to 1986 at the 31 Stations (calculated by monthly volume weighted mean)  ro  0  no trend  -  down trend  +  up trend  Trend of log(S04) from 1980 to 1986 at the 31 Stations (calculated by monthly median)  CO  0  no trend  -  down trend  +  up trend  Trend of log(S04) from 1980 to 1986 in the U S A (calculated by Kriging from monthly volume weighted mean)  -  down trend  +  up trend  Trend of log(S04) from 1980 to 1986 in the U S A (calculated by Kriging from monthly median)  -  down trend  +  up trend  The Locations Of The 81 Monitoring Stations From 1983 To 1986 (For  Sulphate)  Figure 4.2.1  Clustering of S 0 4 monthly volume weighted mean based on sqrt(MSE) " f  1 9 8 3 - 1986  Figure 4.2.2(a)  Clustering of S 0 4 monthly volume weighted mean based on sqrt(MSE) 1 9 8 3 - 1986  (with  outliers)  s  ca a a , a  dm  • Figure 4.2.2(b)  IM  « 1) U N  ID B « ffl B «  «|  Clusters of S 0 4 monthly volume weighted mean based on sqrt(MSE) 1 9 8 3 - 1986 (k=3)  C D  Figure 4.2.3  Histograms of Transformed S 0 4 (Volume Weighted Mean, 83-86) Histgram of S04 (volume weighted mean, 83 - 86)  Histgram of log(S04) (volume weighted mean, 83 - 86)  IL oo o Histgram of sqrt(S04) (volume weighted mean, 83 - 86)  Histgram of (S04) (1/4) (volume weighted mean, 83 - 86) A  0.0  Figure 4.2.4  0.5  1.0  1.5  2.0  2.5  3.0  Histograms of log(S04) by Clusters (Volume Weighted Mean, 83-86) Histogram (or Cluster 1 of log(S04) (vwm, 83 - 86) based on S04  Histogram for Cluster 2 of log(S04) (vwm, 83 - 86) based on § 0 4  Histogram for Cluster 3 of log(S04) (vwm, 83 - 86) based on S 0 4  LL, Figure 4.2.5  Yearly Effect of log(S04) for 3 Clusters (monthly volume weighted mean, 1983 - 1986)  1983  1984  1985 Figure 4.2.6  82  1986  Yearly Effect of log(S04) for 3 Clusters (monthly median, 1983 - 1986)  J  1983  1984  1985 Figure 4.2.7  83  .  u 1986  Monthly Effect of log(S04) for 3 Clusters (monthly volume weighted mean, 1983 - 1986)  ^ o  §  I  ll 1 Jan  1  1  1 Mar  I  1  1  May  I 1  I  I  1  Jul  Figure 4.2.8  84  I  I  Sep  1  2 Nov  Jan  Monthly Effect of log(S04) for 3 Clusters (monthly median, 1983 - 1986)  1 Jan  1 Mar  1 May  1 Jul Figure 4.2.9  85  1  Sep  1 Nov  2 Jan  Station Effect of log(S04) for Cluster 1 (monthly median/volume weighted mean (sorted by volume weighted mean), 1983 -1986)  CO d  Common Effect for Mean Common Effect for Median =  CM  0.961 1.026  d  o d  oo  &  CD  C  CD  o  •3  CM  9 CO  9  d  co co  CD T -  co co  co r-  O  O  iv jt  co o>  TlCM  co co  CO  O  co in CO CM  co  in fO  T- to iT- T- CoM co  IV  co  co  co  o CM o  co  CM  CO o  co  t CO o  CO CO  •0-  O  LO  d  stations  Figure 4.2.10  JO  IO (D O  CO CM hCM  CO co  (0 10  CO LO o o  CO O TT  CO T— Tt  CO LO CM  CO CO LO  CO  T-  CM o  CO  CD  ino  Station Effect of log(S04) for Cluster 2 (monthly median/volume weighted mean (sorted by volume weighted mean), 1983 - 1986)  co o  Common Effect for Mean = Common Effect for Median =  0.528 0.590  eg o  7 j  n  /  \ CO  CO  C  o  o  d  -<  / / / / / /  CM  / u  \  /  \  HN / N  \  Ld  \ \  modian  mean  \ \  /  \ CQ  CM in o  cu O) CO o  CO y-  O  a  in CM  eg CO in o  co O) o  co CO eg o  stations  Figure 4.2.11  CO eg  Station Effect of log(S04) for Cluster 3 (monthly median/volume weighted mean (sorted by volume weighted mean), 1983 - 1986) Common Effect for Mean = 0.083 Common Effect for Median = 0.137  I P P  WW fPTT  1  MM median  trj LO  mtan  trj nj rrj nj CO  CO  ( I l N N N i n O l f t W N r N l D r - N N l f J r o t D ^ f D W ^ N O N n n i n O O C V 1 0 0 0 0 W ^ O ^ O O O C \ I C V J O J - r - 0 - i - O C O C \ I O C v J O O C N J O C \ I O C J O O - » - C \ I O C \ | O C \ i O  stations  Figure 4.2.12  Boxplot for the resid. of log(S04) (83-86, monthly volume weighted mean, clust 1)  oo CD  L X f J -  o CO  -p "  l  •  co  O CM  o  1  •  co co  i-  o  i  i  I L  T  i  Bi  1  CO C M CM  o  "-p  i  1  1  CO CM CO o  CO CO CO  1  !  I  f I  '  I I JL  !  1  I  CO  in  o  CO  sO  CD O C OC t  T•3o  o  t5 Tt  o  co co  LO  ro  CO to  o  station  Figure 4.2.13(a)  to  Tf  CO  xi  c o c o c o c o c o c o c o c o O L O * - C O T - C n C M L O ^ r v c o c D i v T r r ^ c o  LO Cl COO  O  T  -  T  -  ^  C  M  C  M  C  M  co o  Boxplot for the resid. of log(S04) (83-86, monthly median, clust 1)  co o  i !  i i  I  i i  i  • i  j  I  f  !  i  T  i  i  i  o CO  o  .  W c o o ^ O  O  n c O  « i i  •  i  i  j t o i O r t n j W f t i c o n j R i o J w n j c o ^ o ^ - m c j c o o T - t o t ^ m c o c o c o ^ i n c O  O  O  O  O  O  O  C  3  O  O  O  O  O  O  O  O  T  -  station  Figure 4.2.13(b)  T  -  i  o T  i i  T  (- Q - t - o -c -o c o m m ^c Q t o n ) o u > T - c o « - C 3 > c y m o i O T - c O ' - g  c  -  C  \  I  C  \  J  C  \  I  8  Boxplot for the resid. of log(S04) (83-86, monthly volume weighted mean, clust 2)  Figure 4.2.14(a)  Boxplot for the resid. of log(S04) (83-86, monthly median, clust 2)  station  Figure 4.2.14(b)  Boxplot for the resid. of log(S04) (83-86, monthly volume weighted mean, clust 3) CO  I  f  I c o w c o r a n j c t j c o n j r a r a w r a c o c o c a ^ O r W N c o o v i f l i D N r a o i ^ c o o ^ ^ rz ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ O O O O O O O O O O O O O O O O O O O O O T -  c o r a c s c B w r a o i - ^ t o s c o o n t D w n w o ' i i n m T - n i f l c o p g ^ Q f f i ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ ^ . ^. \P. V. tr, tr. tr, tr. tr. 99, ^, ??• » - - r - ^ T - C J C S I C \ ( C M C J C \ J C \ I C M C \ I C J C V 1 0 J C M C 0 C 0 m  ,  station Figure 4.2.15(a)  ^  Boxplot for the resid. of log(S04) (83-86, monthly median, clust 3)  I  I  !  '!!i! i  ! ' ! "  i  Ti  i ! i  T T ! i .! i T l T ' i i i T ! ! 1  1  1  •1  c O r t c d n j r a r o r t N n < i o i on i i ont io r \ l ( M ( O O O O O O O O O O O O O O O O O O O O O T - T - - ^ T - T -  w \JI %AJ  T f O T - c g c ^ o ^ p n L o c p r ^ c p c n - r - c p P T - ' r  Ul U  ^1  U/ W  1 ^  ^ W  n o N n t** OO — — -  station Figure 4.2.15(b)  ..  n j r u r C f O c o t B r a t u r d r t r d i Q O ' - n i f l o B O i - N O ) « r >S rS> Sr N > "f f l — it io ii n ii -n. N i n— f f— l S — i o M C V I C M N N C M C M N ( M < M r i i n c O  Summary of the Effects and Residuals from Median Polish of log(S04) (83-86,monthly volume weighted mean,clust 1) •  -  • •  -  1 1  T  1  T  E  1  1 1  • •  -  t  1  T  '  • •  \  1 1  1 t  •  • Mon  Yr  Sit  MonYr  Figure 4.2.16(a)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(S04) (83-86,monthly median,clust 1)  i  i  Mon  Yr  Sit  MonYr Figure 4.2.16(b)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(S04) (83-86,monthly volume weighted mean,clust 2)  I * i  4-  -L  Mon  Yr  Sit  MonYr Figure 4.2.17(a)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(S04) (83-86,monthly median,clust 2)  Sit  MonYr Figure 4.2.17(b)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(S04) (83-86,monthly volume weighted mean.clust 3)  i T  +  I  Mon  Yr  Sit  MonYr Figure  4.2.18(a)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(S04) (83-86,monthly median,clust 3)  T  ! I  i  Mon  Yr  Sit  MonYr  Figure 4.2.18(b)  MonSit  YrSit  res  Trend of log(S04) from 1983 to 1986 at the 81 Stations (calculated by monthly volume weighted mean)  o  0  no trend  -  down trend  +  up trend  Trend of log(S04) from 1983 to 1986 at the 81 Stations (calculated by monthly median)  Figure 4.2.19(b)  0  no trend  -  down trend  +  up trend  Trend of log(S04) from 1983 to 1986 in the U S A (calculated by Kriging from monthly volume weighted mean)  -  down trend  +  up trend  Trend of log(S04) from 1983 to 1986 in the U S A (calculated by Kriging from monthly median)  -  down trend  +  up trend  TABLE 4.1.1 RESULTS OF THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF SULPHATE (Monthly Volume Weighted Mean, ' 8 0 - ' 8 6 ) site ID  Z  004a 011a 017a 020a 021a 022a 023a 030a 031a 032a 034a 036a 038a 039a 040a 041a 049a 051a 052a 053a 055a 056a 058a 059a 064a 074a 075a 076a 168a 171a 173a  -3 .047 -3 .303 -0 .750 -2 .516 -0 . 958 -2 .222 -2 .391 -2 .381 -2 .728 -2 .828 -2 .104 -0 .761 -2 .797 -2 . 906 -1 .583 -1 .398 -1 . 468 -3 .851 -2 .048 -1 . 445 -1 . 919 -0 . 999 0 .213 -2 .234 -1 .141 -1 .758 -0 .936 -3 .831 -2 . 610 -1 .736 -4 .274  p-value 0 .002 0 .001 0 . 453 0 .012 0 .338 0 .026 0 .017 0 .017 0 .006 0 .005 0 .035 0 .447 0 .005 0 .004 0 .113 0 .162 0 .142 0 .0 0 .041 0 .148 0 .055 0 .318 0 .831 0 .025 0 .254 0 .079 0 .349 0 .0 0 .009 0 .083 0 .0  est'd slope  L--bd  -0..005 -0..007 -0.. 001 -0..003 -0..001 -0..003 -0..003 -0..004 -0..004 -0,.003 -0..003 -0..001 -0..005 -0 .004 . -0..002 -0..002 -0 .002 , -0 .007 . -0..004 -0 .003 . -0 .002 , -0,.001 0..0 -0..003 -0..001 -0..004 -0..001 -0..006 -0..004 -0..002 -0..009  -0 .007 -0 .010 -0 . 004 -0 .004 -0 .003 -0 .005 -0 .005 -0 .007 -0 .006 -0 .005 -0 .005 -0 .003 -0 .008 -0 .006 -0 .004 -0 .004 -0 .005 -0 .009 -0 .006 -0 .005 -0 . 003 -0 .002 -0 .001 -0 .006 -0 .003 -0 .007 -0 .003 -0 .008 -0 .006 -0 .003 -0 .012  1 05  80 %  U'-bd  -0 .003 -0 .004 0 .001 -0 .001 0 .0 -0 .001 -0 .001 -0 .002 -0 .002 -0 .002 -0 .001 0 .001 -0 .003 -0 . 002 0 .0 0 .0 0 .0 -0 .004 -0 .001 0 .0 0 .0 0 .0 0 . 001 -0 .001 0 .0 -0 .001 0 .0 -0 .004 -0 .002 0 .0 -0 .007  RESULTS OF THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF SULPHATE (Monthly Median, '80 - '86) site ID 004a 011a 017a 020a 021a 022a 023a 030a 031a 032a 034a 036a 038a 039a 040a 041a 049a 051a 052a 053a 055a 056a 058a 059a 064a 074a 075a 076a 168a 171a 173a  Z -2 -2 -1 -2 -0 -3 -2 -2 -3 -1 -2 -0 -2 -0 -1 -0 -3 -4 -1 -1 -1 -0 -0 -1 -2 -2 -0 -3 -2 -1 -3  p-value  .445 .728 . 148 . 107 .541 .099 .000 . 478 .286 . 957 .420 .715 . 982 .646 .857 .639 .494 .306 .447 .070 .304 .840 .786 .498 .240 .479 .387 . 608 .601 .639 .443  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  .014 .006 .251 .035 .588 .002 .045 .013 .001 .050 .016 . 475 .003 .518 .063 .523 .0 .0 .148 .285 .192 . 401 . 432 .134 .025 .013 .699 . 0 .009 .101 .001  est'd slope  L-bd  -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 -0 -0 -0 -0  -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0  .004 .006 .002 .003 .001 .005 .004 .004 .006 .002 .004 .001 .005 .0 .003 .001 .006 .007 .003 .002 .001 .001 .001 .003 .002 .005 .0 .006 .004 .002 .007  1 06  80%  .006 .009 .004 .005 .002 .007 .007 .006 .008 .004 .006 .004 .009 .003 .005 .002 .009 .009 .005 .005 .003 .003 .002 .005 .004 .009 .002 .008 .006 .004 .010  U-bd - 0 . .001 - 0 .. 0 0 3 0 ..0 -0. .001 0 .. 0 0 1 -0. .003 - 0 .. 0 0 1 -0. .002 -0. .003 -0. .001 - 0 ,. 0 0 2 0 ,. 0 0 1 -0. .003 0 .. 0 0 1 0 .. 0 0 .. 0 0 1 - 0 . .004 - 0 . .005 0 ..0 0 .. 0 0 .. 0 0 .. 0 0 ..0 0 .. 0 - 0 . .001 - 0 . .002 0 .. 0 0 1 - 0 . .004 - 0 . .002 0 ..0 - 0 . .005  TABLE 4 . 2 . 1 RESULTS OF THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF SULPHATE (Monthly Volume Weighted Mean, ' 8 3 - ' 8 6 ) site ID 004a 010a 011a 012a 017a 020a 021a 022a 023a 024a 025a 028a 029a 030a 031a 032a 033a 034a 035a 036a 037a 038a 039a 040a 041a 046a 047a 049a 051a 052a 053a 055a 056a 058a 059a 061a 063a 064a 065b 068a 070a 071a 073a  Z -3 -1 -1 -0 2 0 1 0 1 -1 0 1 -2 0 -0 0 -0 -2 -1 1 -1 -0 0 1 1 0 1 -0 0 -0 1 1 1 1 -1 -1 1 1 1 -1 -0 -0 2  p-value  . 045 .247 .573 . 460 . 137 . 605 .200 .098 .164 .040 .524 . 654 .053 . 624 .752 .028 .498 .102 .550 . 831 .723 .138 .169 .395 .040 . 667 .173 .160 .660 .786 .733 .164 .752 .786 .247 .810 .458 .067 . 194 .032 .492 . 667 .055  0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  .002 .212 .116 . 646 .033 .545 .230 .922 .244 .298 .600 .098 .040 .533 .452 . 978 . 619 .036 .121 .067 .085 .891 .866 .163 .298 .505 .241 .873 .509 .432 .083 .244 .080 .074 .212 .070 .145 .286 .232 .302 . 623 .505 .040  est'd slope  L-bd  -0 -0 -0 -0 0 0 0 0 0 -0 0 0 -0 0 -0 0 -0 -0 -0 0 -0 0 0 0 0 0 0 -0 0 -0 0 0 0 0 -0 -0 0 0 0 -0 -0 -0 0  -0 -0 -0 -0 0 -0 -0 -0 0 -0 -0 0 -0 -0 -0 -0 -0 -0 -0 0 -0 -0 -0 0 -0 -0 0 -0 -0 -0 0 0 0 0 -0 -0 0 -0 0 -0 -0 -0 0  . 009 .006 .009 .002 .007 .001 . 003 .001 .003 .003 .001 .004 .006 .002 .003 .0 .002 .007 .010 .004 .007 .0 .001 .004 .003 .002 .004 .001 .002 .003 .006 .002 .005 .005 .004 .009 .003 .003 .003 .007 .002 .003 .007  107  80%  .013 .012 .014 .011 .002 .002 .001 .004 .0 .00 6 .002 .001 .011 .003 .008 .004 .007 .011 .014 .002 .012 .006 .004 .0 .001 .002 .0 .006 .002 .008 .002 .0 .001 .002 .008 .014 .0 .001 .0 . 014 .008 .008 .002  U-bd - 0 .. 0 0 6 0 .. 0 -0. .001 0 .. 0 0 3 0 .. 0 1 2 0 .. 0 0 5 0 .. 0 0 7 0 .. 0 0 4 0 .. 0 0 7 0 .. 0 0 1 0 .. 0 0 4 0 .. 0 1 0 -0. .002 0 .. 0 0 7 0 .. 0 0 3 0 ,. 0 0 3 0 ,. 0 0 2 -0, .002 - 0 ,. 0 0 2 0 .. 0 0 8 - 0 ,. 0 0 2 0 ,. 0 0 4 0 ,. 0 0 6 0 ,. 0 0 7 0 ,. 0 0 7 0 .. 0 0 6 0 ,. 0 0 7 0 ,. 0 0 5 0 .. 0 0 7 0 ,. 0 0 2 0 ,. 0 1 0 0 ,. 0 0 5 0 .. 0 0 8 0 .. 0 0 8 0 ,. 0 -0. .003 0 .. 0 0 5 0 .. 0 0 6 0 .. 0 0 6 0 .. 0 0 2 0 .. 0 0 4 0 .. 0 0 3 0 .. 0 1 2  TABLE 4.2.1  (continued)  RESULTS OF THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF SULPHATE (Monthly Volume Weighted Mean, '83 - '86) site ID 074a 075a 076a 077a 078a 160a 161a 163a 164a 166a 168a 171a 172a 173a 249a 251a 252a 253a 254a 255a 257a 258a 268a 271a 272a 273a 275a 277a 278a 279a 280a 281a 282a 283a 285a 349a 350a 354a  Z  p-value  -1 .311 4 .035 -1 .211 0 .0 -3 .025 -3 .278 0 .116 0 .038 -0 .956 -1 .229 0 .294 1 .591 -2 . 994 -2 . 427 1 . 644 1 .360 -2 . 471 0 .293 -1 .468 -2 .578 0 .415 0 .0 1 .244 -1 .477 0 .436 -0 . 924 0 .836 2 .384 -2 . 641 -1 .324 -1 .751 1 .816 1 . 635 -1 .577 1 .401 1 . 420 2 .113 -3 .539  0.190 0.0 0.226 1.000 0.002 0.001 0.908 0.970 0.339 0.219 0.769 0.112 0.003 0.015 0.100 0.174 0.013 0.769 0.142 0.010 0.678 1.000 0.213 0.140 0.663 0.355 0.403 0.017 0.008 0.185 0.080 0.069 0.102 0.115 0.161 0.156 0.035 0.0  est'd slope  L-bd  -0.007 0.012 -0.003 0.0 -0.011 -0.013 0.001 0.0 -0.004 -0.004 0.001 0.004 -0.011 -0.008 0.007 0.005 -0.008 0.001 -0.005 -0.007 0.001 0.0 0.003 -0.005 0.001 -0.002 0.003 0.008 -0.008 -0.007 -0.006 0.010 0.005 -0.004 0.006 0.003 0.010 -0.018  -0 . 013 0 .008 -0 .008 -0 .005 -0 .015 -0 .019 -0 .004 -0 .004 -0 .009 -0 .008 -0 .003 0 . 001 -0 .016 -0 .012 0 .001 0. 0 -0 .010 -0 .003 -0 .010 -0 .012 -0 .003 -0 . 004 0 .0 -0 .009 -0 .001 -0 .006 -0 .002 0 .004 -0 .014 -0 .011 -0 .010 0 .002 0 .001 -0 .008 0 .0 0 .0 0 .004 -0 .025  1 08  80% U-bd 0 .0 0 .016 0. 0 0. 004 -0. 007 -0. 008 0. 005 0 .005 0. 001 0. 0 0 .004 0. 008 -0 .006 -0 .004 0. 015 0 .011 -0 .004 0. 004 -0 .001 -0. 004 0. 008 0. 004 0 .009 0 .0 0. 003 0. 001 0. 007 0. 011 -0. 004 0. 0 -0. 002 0. 018 0. 009 -0. 001 0. 013 0. 007 0. 016 -0. 012  TABLE 4.2.2 RESULTS OF THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF SULPHATE (Monthly Medain, ' 8 3 - ' 8 6 ) site ID 004a 010a 011a 012a 017a 020a 021a 022a 023a 024a 025a 028a 029a 030a 031a 032a 033a 034a 035a 036a 037a 038a 039a 040a 041a 046a 047a 049a 051a 052a 053a 055a 056a 058a 059a 061a 063a 064a 065b 068a 070a 071a 073a  Z -2 .531 -1,.385 - 1 ..209 -0, .029 1,. 8 6 5 0 ,. 0 3 7 0 ,. 6 8 4 -0, .836 0 ,. 9 3 3 - 1 ,.173 -0. .133 0 .. 3 3 3 - 1 ..680 0 .. 3 4 9 - 1 ,. 082 - 1 ,. 082 -0, .560 - 2 .. 5 0 9 - 1 ..246 1.. 2 6 2 - 2 .. 1 6 0 0 .. 1 4 7 1,. 0 7 6 0 .. 3 5 6 1.. 1 1 1 - 0 .. 0 6 2 0 .. 8 2 9 0 .. 3 2 0 0 .. 8 6 2 0 .. 9 8 5 1.. 4 8 7 1,. 6 4 4 1.. 2 3 5 2 .. 5 0 7 -0. .541 - 1 ..576 0 .. 4 6 8 0 .. 7 2 0 0 .. 0 0 .. 0 4 7 - 1 ..036 - 1 ..182 2 .. 1 4 3  p-value 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0  .011 .166 .227 . 977 .062 . 971 .494 .403 .351 .241 .894 .739 .093 .727 .279 .279 .576 .012 .213 .207 .031 .883 .282 .722 .267 . 950 . 407 .749 .389 .325 .137 .100 .217 .012 .588 .115 . 640 .472 .000 .962 .300 .237 .032  est'd slope  L-t'd  -0 -0 -0 0 0 0 0 -0 0 -0 0 0 -0 0 -0 -0 -0 -0 -0 0 -0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0 -0 0 0 0 0 -0 -0 0  -0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 -0 -0 0 -0 0 -0 -0 -0 -0 -0 0 0 0 0 -0 -0 -0 -0 -0 -0 -0 -0 0  .008 .004 .007 .0 .007 .0 .002 .003 .004 .004 .0 .001 .008 .002 .005 .004 .002 .009 .006 .004 .005 .0 .005 .001 .003 .0 .003 .002 .003 .004 .006 .004 .004 .005 .002 .006 .001 .002 .0 .0 .005 .006 .009  109  80%  .012 .011 .014 .008 .001 .003 .002 .008 .002 .008 .005 .004 .013 .003 .011 .011 .006 .013 .013 .0 .010 .003 .0 .003 .0 .005 .001 .005 .002 .001 .001 .001 .0 .002 .006 .012 .003 .002 .004 .005 .012 .012 .004  U-bd -0 0 0 0 0 0 0 0 0 0 0 0 -0 0 0 0 0 -0 0 0 -0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0 0 0 0 0 0 0 0  .004 .0 .0 .005 .012 .003 .006 .001 .009 .0 .003 .007 .002 .007 .001 .001 . 002 .004 .0 .007 .002 .004 .010 .005 .005 .004 .007 .007 .007 .009 .012 .007 .008 .008 .003 .001 . 004 .006 .003 .006 .0 .001 .015  TABLE 4.2.2  (continued)  RESULTS OF THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF SULPHATE (Monthly Medain, '83 - '86) site ID 074a 075a 076a 077a 078a 160a 161a 163a 164a 166a 168a 171a 172a 173a 249a 251a 252a 253a 254a 255a 257a 258a 268a 271a 272a 273a 275a 277a 278a 279a 280a 281a 282a 283a 285a 349a 350a 354a  Z  p--value  -1 . 954 3 .760 -2 .100 0 .165 -1 .409 -3 . 992 0 .213 -0 .682 -0 .758 -0 . 935 -0 .161 1 .235 -2 .211 -2 .089 2 .505 1 .702 -1 .902 0 .147 0 .312 -1 .804 0 .526 0 . 104 0 .560 -2 .720 -0 .862 -0 .987 1 . 040 2 . 697 -2 .580 -1 . 440 -1 .218 1 .816 0 .702 -1 .825 1 .808 0 .511 1 .879 -3 .277  0 .051 0 .0 0 .036 0 .869 0 .159 0. 0 0 .831 0 .495 0 .449 0 .350 0 .872 0 .217 0 . 027 0 .037 0 .012 0 .089 0 .057 0 .883 0 .755 0 . 071 0 .599 0 . 917 0 .576 0 .007 0 .389 0 .324 0 .298 0 .007 0 .010 0 .150 0 .223 0 .069 0 .483 0 .068 0 .071 0 .609 0 .060 0 .001  80 %  est'd slope  L-bd  -0. 007 0. 012 -0. 008 0. 001 -0. 007 -0 .019 0. 001 -0. 003 -0. 004 -0. 004 0. 0 0. 003 -0. 006 -0. 008 0 .008 0. 007 -0. 004 0. 0 0. 001 -0. 009 0. 002 0 .0 0. 002 -0. 007 -0 .002 -0. 003 0. 004 0. 007 -0. 010 -0. 006 -0. 005 0. 010 0 .002 -0. 005 0. 008 0. 001 0. 007 -0. 013  -0 .014 0 .008 -0 .013 -0 .004 -0 .015 -0 .028 -0 . 004 -0 .008 -0 .010 -0 . 010 -0 .005 0 .0 -0 .011 -0 .014 0 .003 0 . 001 -0 .008 -0 .005 -0 .006 -0 .015 -0 .003 -0 .005 -0 .004 -0 .011 -0 .005 -0 .007 -0 .001 0 .003 -0 .015 -0 .012 -0 .011 0 .003 -0 .002 -0 .010 0 .002 -0 .002 0 .002 -0 .019  110  U-]bd -0 .003 0 .016 -0 .003 0 .006 0 .0 -0 .012 0 .005 0 .002 0 .003 0 .001 0 .003 0 .007 -0 .003 -0 .004 0 .016 0 .014 -0 .001 ' 0 .005 0 .007 -0 .003 0 .008 0 .005 0 .009 -0 .004 0 .001 0 .001 0 .009 0 .011 -0 .005 -0 .001 0 .0 0 .019 0 .007 -0 .002 0 .013 0 .005 0 .012 -0 .009  Chapter 5  ANALYSES AND CONCLUSIONS FOR NITRATE  Monthly precipitation volume weighted mean and monthly median are are referred to as "weighted mean" and "median", respectively, below.  5.1  Results for the Historical Data "Historical data" refers to the data collected from January, 1980 to December, 1986. For  N O ^ , there are 31 sites with 5 or fewer missing observations during the period. The locations of these 31 sites are plotted in Figure 5.1.1.  From this figure we can see that  more sites are located in the East than the West. The results of this section are based on the data obtained from these 31 sites.  5.1.1  Clustering and Transformation  Before clustering the data, one outlier was deleted from the data set, namely the observation at the site with ID=040a (see September, 1986. at that site. means.  Olsen and Slavich, 1986), observed in  Its value is 37.76, much larger than 6.07, the second largest observation  Figure 5.1.2 (a) shows the hierarchical cluster structure of the weighted  The sites are partitioned quite naturally into two clusters.  sites labelled by cluster are shown in Figure 5.1.3.  The locations of the  The pattern is obvious: cluster 1 is  located in the subregion of highest industrial concentration; and cluster 3 is spread over the rest of the United States. This indicates that the concentration of NO^ in wet deposition is higher for  ^  n d  -  u s t r  ^  a r e a s  t  n  a  n  t n a t  f ° other areas. r  A cluster analysis was done  without deleting outliers. The clusters are the same however, except for site 040a which is ruled out of any cluster (Figure 5.1.2 (b) ). 1 1 1  The  histograms of the original data and the transformed data under the three kinds of  transformations described in Section 3.1, are shown in Figure 5.1.5.  It appears that the  histograms of the log transformed data and the 1/4 power transformed data fit the normal curve best. But they still have long left tails. The other two are right skewed.  Since the  log transformation is good for many other cases as well, it is adopted in this case and the remainder of our analysis of the nitrate data is based on the transformed data.  The  histograms of the log transformed data for each cluster are depicted in Figure 5.1.5. It turns out that both histograms have log left tails and the one for cluster 2 having the heavier tail. In general, neither histogram fits the normal curve well.  5.1.2 The  Trend, Seasonality and Spatial Patterns analysis described below is done for both log transformed monthly volume weighted  mean and log transformed monthly median N O 3  data. The analysis is applied separately to  each cluster obtained from clustering the volume weighted mean data. Three-way median polishes are applied to both weighted means cluster, in the manner described in Section 3.2.  and medians in each  Figures 5.1.6 and 5.1.7 show the plots of  the yearly effects of the two clusters for weighted mean and median, respectively. show similar patterns although the one for the medians seems more stable. show down-trends before 1983.  They  Both clusters  After 1983, the data of cluster 1 show an increasing trend  while that for cluster 2 is basically flat. This suggests that the concentration of NOg  in wet  deposition increased after 1983 for the most industrialized subregion of the United States and had remained constant for the rest of the United States to 1986.  Figure 5.1.8 and Figure 5.1.9 show the monthly effects for the weighted means and medians, respectively.  They have almost the same pattern .  The patterns of the two  clusters are the same as well. The effects are high from January to August and low from  112  September to December.  Sudden jumps occurred around September and December.  addition, there is a gap around March.  In  The fact that the seasonal patterns are very similar  for all three clusters suggests a reassuring spatial stability in this seasonal pattern. Further study of this interesting pattern is needed.  Figures 5.1.10 to 5.1.11 display the site effects of log(NOg) by cluster; the weighted means and medians are plotted together and sorted by weighted means within each cluster. The site effects of Station 041a, 055a and 056a for weighted means in Figure 5.1.10 are quite different from those of the medians.  This may be caused by the fact that the  distributions of the data from these stations are quite different from those of the other stations.  In this figure, the stations located in the center of the industrial area tend to have  large positive effects.  In Figure 5.1.11, Stations 059a and 074a, the only two stations  located along the West Coast, have remarkably large, negative effects.  This fact are  reflected in the structure of clustering analysis as well (see Figure 5.1.2).  Figures 5.1.12 to 5.1.13 are the boxplots of the residuals of the three-way median polish of  the log(NOg) data where (a) is for weighted means and (b) is for medians and each box  represents one site.  In general, the variations of the residuals in each cluster are of the  same order and comparable. The differences between the boxplots for the means and for the medians are even smaller. In Figures 5.1.12 ((a) and (b)), the boxes for Stations 020a and 038a are larger than the others.  In fact, these two stations are the ones farthest from  the center of the industrial area within cluster 1.  In addition, we observe a few extreme  outliers from Station 040a, and from Station 075a and 168a as well. For these sites, further detailed study is needed.  In Figures 5.1.13 ((a), (b)), extreme outliers are observed in the  boxplots for Station 034a, 039a, 049a and 059a and they need a more detailed study as well. But no analyses for specific site are included in this paper. T h e boxes for the two stations  113  on West Coast, 059a and 074a, are relatively large indicating not only that they have large negative effects but also that they behaved differently from the others so that the model can not fit them well. Figures 5.1.14 to 5.1.15 summarize the median polish for the two clusters, respectively where (a) is for the weighted means and (b) is for the medians. represent the main effects, interactions and residuals respectively.  In each figure, boxes We regard interactions  as non-negligible if the sizes of the boxes for the interactions are comparable with those for main effects.  In Figures 5.1.14 (a) and (b), the boxes for yearly effects are relatively small,  especially in (b).  This suggests that the variation caused by trend is smaller than the  variations caused by the other effects.  The variation of the interactions between yearly  effects and site effects is small as well. The two boxes for the interactions of month-byyear and month-by-site are relatively large suggesting that their contributions to the model should be taken into account. The pattern of the summary for cluster 2 is quite similar to that for cluster 1 except the effects of the two sites on the West Coast are ruled out of the box for site effects as two outliers (see Figures 5.1.15 (a) and (b)).  5.1.3  The Results of Trend Testing, Slope Estimation and Kriging  Mann-Kendall Test and Sen's nonparametric slope estimation procedure are applied to the deseasonalized data for each site in the manner described in Section 3.3. Kendall  A Mann-  Test statistic with the corresponding p-value and Sen's (1968b) nonparametric  slope estimate with its 80% confidence interval are obtained for each site and for both weighted means and medians. These are shown in Tables 5.1.1 and 5.1.2 respectively.  In  Table 5.1.1, 14 out of the 31 sites have significant trends at the p=0.2 level. In Table 5.1.2, 15 out of the 31 sites have significant trends at the p=0.2 level.  114  The results of the nonparametric slope estimation are plotted in Figures 5.1.16 (a) and (b) with symbols defined in Section 3.3, for the weighted means and medians, respectively. If the lower 80% confidence bound of the slope estimate is greater than 0 at any site a "+" is plotted; if the upper 80% confidence bound of the slope estimate is less than 0 a "-" is plotted. Otherwise a "0" is plotted . "+", "-" and "0" represent respectively, an up-trend, a down-trend and no-trend. The patterns of these two figures are very similar. More than half of the sites show no-trends and they are mainly located in the Eastern United States. The data from rest of the sites located mainly in the other areas of the United States show a down-trend.  This conclusion agrees with the previous finding that the wet deposition  concentration of N O ^ has no trend or down-trend for most areas of the United States (Schertz and Hirsh,1985). However, as we mentioned before, this could be a misleading conclusion in the presences of a V-shaped trend. At the very least, this result says that the concentrations of nitrate for the latter years of the sampling period are about the same or smaller than that of early years of the same period.  The results of Kriging the slope estimates are displayed in Figures 5.1.17(a) and (b). As before, Figure 5.1.17(a) is for the weighted means and (b) is for medians. A "+" means an up-trend is estimated at the corresponding point and a "-" means a down-trend, estimated at the point. Again, these two plots are similar. Both Figures 5.1.17 (a) and (b) show that there is a no-trend for the Eastern United States and a down-trend for the rest of the United States.  Considering that these results are obtained from the same set of data  which produce Figures 5.1.16 (a) and (b), this is quite natural.  The difference between  these two figures are probably caused by the differences in magnitudes of the slope estimates of weighted means and of medians. Note that these two figures can only be used as a guideline for the spatial distribution of the trends and should be read in conjunction with Figure 5.1.16.  At any particular point where the observations were not made,  interpolation may not be reliable. This is because (a) estimation is based on the estimated  115  slopes, not on the observations directly and (b) the spatial resolution of the measurements is low and therefore, the correlations between sites are small.  5.2  Results for the Recent Data "Recent data" refers to the data collected from January, 1983 to December, 1986. For  N O ^ , there are 81 sites with 5 or fewer missing observations during the period. The locations of these 81 sites are plotted in Figure 5.2.1.  From this figure we can see that  more sites are located in the Northeast than the rest of the areas.  The results of this  section are based on the data obtained from these 81 sites.  5.2.1 Clustering and Transformation Before clustering the weighted mean data, a outlier was deleted from the data set, namely the observation for the Station 040a (see Olsen and Slavich, 1986).  It was  observed in September, 1986, the same one mentioned in Section 5.1.1. Figure 5.2.2 (a) shows the hierarchical cluster structure of the weighted means.  The sites first partition  naturally into two clusters: a large one on the left and a smaller one on the right. If we ignore Stations 035a, 268a and 283a, then the large cluster can be partitioned into two subgroups: cluster 2 and the group of the rest of the stations, as shown in Figure 5.2.2. For technical convenience, Stations 035a, 268a and 283a were assigned to the group closest to these three stations.  We label this group as cluster 1, as shown in Figure 5.2.2.  The  locations of the sites labelled by clusters are shown in Figure 5.2.3. Keeping in mind that clusters 1 and 2 are subgroups of one big cluster, the pattern of Figure 5.2.3 is similar to that of Figure 5.1.3. In fact, cluster 3 corresponds to cluster 1 in the latter, located in the subregion of highest industrial concentration; and clusters 1 and 2 in the former correspond to cluster 2 in the latter, located throughout the rest of the United States.  116  This gives us the  same indications we mentioned in Section 5.1.1, namely, that the concentration of N O ^ in wet deposition is higher for the industrial areas than that for other areas.  A cluster  analysis was done without deleting outliers. The clusters are the same except for Station 040a which is excluded (Figure 5.2.2 (b) ).  The histograms of the original data and the transformed data under the three kinds of transformations described in Section 3.1, are shown in Figure 5.2.5. It appears that the histograms of the log transformed data and the 1/4 power transformed data fit the normal curve best. But they still have long tails. The other two are right skewed. We adopt the log transformation in this case since it is the best transformation in many other cases and, particularly, since it was adopted in the case of "historical data". The remaining analyses of this section are based on the log transformed data. The histograms of the log transformed data for each cluster are shown in Figure 5.2.5. It turns out that all the histograms have long tails while the tail in the data for cluster 1 is heavy. 5.2.2  T r e n d , Seasonality a n d Spatial Patterns  The analysis described below is applied to both log transformed monthly volume weighted means and log transformed monthly medians. The analysis is applied separately to each cluster obtained from clustering the volume weighted mean data. Three-way median polishes are applied to both weighted means and medians in each cluster. Figures 5.2.6 and 5.2.7 show the plots of the yearly effects of the three clusters for weighted mean and median respectively. Combining these two figures, it seems that the sites in cluster 1 reveal a degree of down trend while the trends for clusters 2 and 3 are not obvious. Observe that here cluster 3 and cluster 1 in Figure 5.1.3 cover essentially the same geographic subregion while clusters 1 and 2 correspond to cluster 2 in Figure 5.1.3; the pattern of the trends observed here are consistent with the pattern of the trends in Figures 5.1.6 and 5.1.7.  117  Figures 5.2.8 and 5.2.9 show the monthly effects for the weighted means and medians, respectively. Although more than half of the data used in the present analyses were not in the historical data, the patterns seen here are almost the same as those displayed in Figures 5.1.8 and 5.1.9 for historical case: the effects are high from January to August and low from September to December; two sudden jumps around September and December and a gap around March.  This convinces us again of spatial stability in the seasonal pattern.  Cluster 1 seems have a larger variation than the other clusters.  Figures 5.2.10 to 5.2.12 show the site effects of log(NOg) by cluster; the weighted means and medians are plotted together and sorted by weighted means within each cluster. For cluster 1, the stations located in the middle of the United States have large positive effects while the stations located along the West Coast have large negative effects (see Figure 5.2.10). In cluster 3, the stations close to the center of the industrial region tend to have a positive effects and the stations located far from the center of the industrial region tend to have a negative effects (see Figure 5.2.12).  In general, the closer a station is  located to the center of the industrial region, the more likely it has a large positive effect.  Figures 5.2.13 to 5.2.15 are the boxplots of the residuals of the three-way median polish of the log(NO^) data where (a) is for weighted means and (b) is for medians and each box represents one site.  In general, the variations of the residuals in each cluster are of the  same order and are comparable to each other and to those in Figures 5.1.12 to 5.1.13 as well.  The differences between the boxplots for the means and for the medians are even  smaller. Extreme outliers are found in Stations 035a, 074a, 078a and 281a in Figures 5.2.13 (a) or (b), 034a, 039a, 049a and 349a in Figures 5.2.14(a) or (b) and 040a, 075a,168a and 350a in Figures 5.2.15 (a) or (b).  Among these stations, 034a, 039a, 040a, 049a, 075a,  168a, have the extreme outliers observed in Figures 5.2.12 to 5.1.13 and those outliers may  118  be the same.  Some other stations do have oudiers as well.  And all these stations need  further study. Figures 5.2.16 to 5.2.18 summarize the median polish for the three clusters respectively where (a) is for weighted means and (b) is for medians. In each figure, boxes represent the main effects, interaction and residuals.  In Figures 5.2.16 (a) and (b), we note that the  variation of the station effects are quite large. The reason for this is that the Station 035a, 268a and 283a have large positive effects, which can be seen from Figure 5.2.10. Actually these 3 stations were assigned to this cluster only for technical convenience (see Figure 5.2.2(a)). This causes the large variation of the station effects in this cluster. Among the other effects, the variations in year effects and the year-by-station interactions are relatively small as in the situation portrayed in Figure 5.1.15 (a) and (b). For cluster 2, the variation in the yearly effects and the interactions of yearly effects with other effects are all small.  This suggests that for the stations in this cluster there was little year to year  variation in their data patterns (see Figures 5.2.17 (a) and (b)). show the summaries for cluster 3.  Figures 5.2.18 (a) and (b)  The pattern in this figure is similar to that in Figure  5.1.14.  5.2.3  T h e Results of T r e n d Testing, Slope Estimation a n d K r i g i n g  The Mann-Kendall Test and Sen's slope estimation procedure which were applied to "historical data" are used for "recent data" as well. A Mann-Kendall test statistic with the corresponding p-value and Sen's (1968b) nonparametric slope estimate with its 80% confidence interval were obtained for each site for both the weighted means and medians. These are shown in Tables 5.2.1 and 5.2.2 respectively.  In Table 5.2.1, 28 out of the 81  sites have significant trends at the p=0.2 level. In Table 5.2.2, 27 out of the 81 sites have significant trends at the p=0.2 level.  119  Figures 5.2.19 (a) and (b) are obtained by the same method used to produce Figures 5.1.16 for weighted means and medians respectively. Again, "+", "-" and "0" represent an up-trend, a down-trend and no-trend, respectively.  In Figure 5.2.19 (a), 12 of the sites  show up-trends and are mainly located in the Eastern United States.  15 of the sites show  down-trend and are located mainly in the West and middle of the United States. The rest of the sites show no-trend and are located throughout the United States.  The pattern of  Figure 5.2.19 (b) is similar to that in Figure 5.2.19 (a) with fewer "up-trend" stations. This probably is due to the fact that the estimates based on medians are typically more conservative than those based on means.  Comparing Figures 5.2.19 ((a) and (b)) with  Figures 5.1.16 ((a) and (b)), we can see that some of the "-'s" and "0's" in former become "0's" and "+'s" in latter and such changes happened mainly in the Eastern United States. This suggests that although the trends for the stations located in the Western United States continued to decline, most of the stations located in the Eastern United States no longer have down-trends and some of them had an up-trend during 1983 to 1986.  This  implies that some of the stations do have a V-shaped trend.  The results of Kriging the slope estimates are displayed in Figures 5.2.20 (a) and (b) where (a) is for the weighted means and (b) is for the medians. As before, a "+" means an up-trend and a "-" means a down-trend estimated at the corresponding point, respectively. Again, these two plots are similar except that both the "+" area and "-" areas in (b) are smaller than those in (a). Both Figures 5.2.20 (a) and (b) show that there is a down-trend for some areas in the Western United States, an up-trend in the East and no-trend in the rest of the subregions. Comparing these two figures with Figures 5.1.17 ((a) and (b)), we can see that the areas with "-" shrink significantly towards the West and the areas with "+" appear in the East. The difference in the patterns of the trends is quite remarkable. For the reason given in Section 5.1.3, these two figures can only be used as indicators of the spatial distribution of the trends as the interpolation error for a specific point could be large.  120  The Locations Of The 31 Monitoring Stations From 1980 To 1986 (For Nitrate)  Figure 5.1.1  Clustering of N03 monthly volume weighted mean based on sqrt(MSE) 1980- 1986  8  8  |  S  $  a  S  in  R  a e  Figure 5.1.2(a)  r»  Clustering of N03 monthly volume weighted mean based on sqrt(MSE) 1980- 1986 (with outliers)  CO « tn  a  cj  Figure 5.1.2(b)  Clusters of N 0 3 monthly volume weighted mean based on sqrt(MSE) 1980 - 1986 (k=2)  Figure 5.1.3  Histograms of T r a n s f o r m e d N 0 3 (Volume Weighted M e a n , 80-86) Histgram of N03 (volume weighted mean, 80 - 86)  Histgram of bg(N03) (volume weighted mean, 80 - 86)  Histgram of sqrt(N03) (volume weighted mean, 80 - 86)  Histgram of (N03) (1/4) (volume weighted mean, 80 - 86)  ro cn  Figure 5.1.4  A  Histograms of l o g ( N 0 3 ) by Clusters (Volume Weighted Mean, 80-86) Histogram for Cluster 1 of log(N03) (vwm, 80 - 86) based on N03 —  Histogram for Cluster 2 of log(N03) (vwm, 80 - 86) based on N03  ro CD  Figure 5.1.5  Yearly Effect of log(N03) for 2 Clusters (monthly volume weighted mean, 1980 - 1986)  1 27  Yearly Effect of log(N03) for 2 Clusters (monthly median, 1980 - 1986)  980  1981  1982  1983 Figure 5.1.7  128  1984  1985  1986  Monthly Effect of log(N03) for 2 Clusters (monthly volume weighted mean, 1980 - 1986)  1  Jan  1  Mar  1  May  1  1  Jul Figure  Sep 5.1.8  129  1  Nov  2 Jan  Monthly Effect of log(N03) for 2 Clusters (monthly median, 1980 - 1986)  1 Mar  1  May  1 Jul Figure 5.1.9  130  1 Sep  1 Nov  2 Jan  Station Effect of log(N03) for Cluster 1 (monthly median/volume weighted mean (sorted by volume weighted mean), 1980 Common Effect for Mean = Common Effect for Median =  0.609 0.719  7~  / \ / \ / \  7—  / / / / /  \  7K  T  \  /\  /  \  /\  /  \  /\  /  \  /\  /  \  / \  \  /\\ /  / /  /  /  /  /  /  /  \  7 / / \  /  / \ / \ / \  \ \  Xl\  / S  \  / \ / \ / / z  median  moan  /  / CO CO CO  1986)  co eg  co to in  stations  Figure 5.1.10  co in in  3 o  CO  5  CO CD  Station Effect of log(N03) for Cluster 2 (monthly median/volume weighted mean (sorted by volume weighted mean), 1980 -1986) Common Effect for Mean = 0.013 Common Effect for Median = 0.189  n  r - R  nn \h  nR  ffl nH  ftjzi  I  s  I  s  Ms median  A3 Ol LO  CO I'-  to  ll) CO  CO  stations  Figure 5.1.11  CO  CO  RI o  mean  Boxplot for the resid. of log(N03) (80-86, monthly volume weighted mean, clust 1)  station  Figure 5.1.12(a)  Boxplot for the resid. of log(N03) (80-86, monthly median, clust 1)  station  Figure 5.1.12(b)  Boxplot for the resid. of log(N03) (80-86, monthly volume weighted mean, clust 2)  I  eg o  {  rs o  I  }  f  1  !  j  1  eg  rg co o  rg o o  CO  SI o  eg  T  ;  T  eg co co o  1  co cn eo o  i  l »  CO  cn  s  station  Figure 5.1.13(a)  ! 1  I  eg eo m  co co to  V  eg cn m o  |  i  •  i  eg  eg  eg  eg  Boxplot for the resid. of log(N03) (80-86, monthly median, clust 2)  station  Figure 5.1.13(b)  Summary of the Effects and Residuals from Median Polish of log(N03) (80-86,monthly volume weighted mean.clust 1)  Sit  MonYr Figure 5.1.14(a)  MonSit  Summary of the Effects and Residuals from Median Polish of log(N03) (80-86,monthly median,clust 1)  i  Mon  Yr  Sit  MonYr Figure 5.1.14(b)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(N03) (80-86,monthly volume weighted mean,clust 2)  I I  x _J_  Mon  Yr  Sit  MonYr Figure  5.1.15(a)  T  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(N03) (80-86,monthly median,clust 2)  t •  Mon  Yr  Sit  MonYr Figure 5.1.15(b)  MonSit  YrSit  res  Trend of log(N03) from 1980 to 1986 at the 31 Stations (calculated by monthly volume weighted mean)  0  no trend  -  down trend  +  up trend  Trend of log(N03) from 1980 to 1986 at the 31 Stations (calculated by monthly median)  ro  0  no trend  -  down trend  +  up trend  Trend of log(N03) from 1980 to 1986 in the USA (calculated by Kriging from monthly volume weighted mean)  CO  -  down trend  +  up trend  Trend of log(N03) from 1980 to 1986 in the USA (calculated by Kriging from monthly median)  -  down trend  +  up trend  The Locations Of The 81 Monitoring Stations From 1983 To 1986 (For Nitrate)  Figure 5.2.1  Clustering of N03 monthly volume weighted mean based on sqrt(MSE) 1983- 1986  fr g jn 88  3  llii  & K J  s  s  8Ki  8t  Figure 5.2.2(a)  3  v a o •-  cm  o °"  fl  [A  Clustering of N 0 3 monthly volume weighted mean based on sqrt(MSE) 1983- 1986 (with outliers)  A?  rl  rk. O CD co  — CO CJ (O  i ro a  3»  ^ ™ « IB  a  w _,  rVT  ra  r*> I-  83  s  Figure 5.2.2(b)  IB  8  IB  £ m c ° o r  o nn  O O ID  o  r— o  Clusters of N03 monthly volume weighted mean based on sqrt(MSE) 1983 - 1986 (k=3)  Figure 5.2.3  Histograms of Transformed N 0 3 (Volume Weighted M e a n , 83-86) Histgram of N03 (volume weighted mean, 83 - 86)  Histgram of log(N03) (volume weighted mean, 83 - 86)  d3  Histgram of sqrt(N03) (volume weighted mean, 83 - 86)  Histgram ol (N03) (1/4) (volume weighted mean, 83 - 86) A  rrrT 2.0  Figure 5.2.4  2.5  H i s t o g r a m s of l o g ( N 0 3 ) by C l u s t e r s (Volume Weighted M e a n , 83-86) 8  Histogram for Cluster 1 of log(N03) (vwm, 83 - 86) based on N03  Histogram for Cluster 2 of log(N03) (vwm, 83 - 86) based on N03  Histogram for Cluster 3 of log(N03) (vwm, 83 - 86) based on N03  x.t  Figure 5.2.5  Y e a r l y Effect of log(N03) for 3 C l u s t e r s (monthly v o l u m e w e i g h t e d m e a n , 1983 - 1986)  li  L  1983  1984  ]  1985 Figure 5.2.6  151  ij  1986  Yearly Effect of log(N03) for 3 Clusters (monthly median, 1983 - 1986)  u  1983  1984  1985  Figure 5.2.7  152  1986  M o n t h l y Effect of l o g ( N 0 3 ) for 3 C l u s t e r s (monthly v o l u m e w e i g h t e d m e a n , 1 9 8 3 - 1986)  d  u 1  Jan  '  1 1  Mar  1  1  i  1  1  May  i Jul  Figure 5.2.8  1 53  i  i 1  Sep  I  1  Nov  2  Jan  M o n t h l y Effect of log(N03) for 3 C l u s t e r s (monthly m e d i a n , 1983 - 1986)  1  u  1  Jan  1  1  1  Mar  1  1  I  1  1  May  I Jul  Figure 5.2.9  1 54  i  I  1  Sep  i  1  1  Nov  i  2  Jan  Station Effect of log(N03) for Cluster 1 (monthly median/volume weighted mean (sorted by volume weighted mean), 1983 - 1986) Common Effect for Mean = -1.116 Common Effect for Median = -0.964  \ /  / / / / / / / / /  \ \ \ \ \  TK 7 \ / \ /\ / \ /\ /\ /  / \ / \ / \ / \  /\ /U  median  mean  \ \ \ \ 10  10 O)  in o  cd  o  (0 CO  o  T —'  stations  Figure 5.2.10  (0 CO  (fl  CO CO  Station Effect of log(N03) for Cluster 2 (monthly median/volume weighted mean (sorted by volume weighted mean), 1983 - 1986) Common Effect for Mean = -0.084 Common Effect for Median = 0.084  P| H  1  III  n  median  mean  po)tootooiooN^otowo)^nt-oitoNcoioc)No»-^(ji^S(Ot-n[Vi-oiT-toa)So)io O  W  O  T  -  O  x  -  C  y  C  \  J  O  C  A  I  O  O  J  O  r  t  T  -  ^  O  O  T  -  W  O  C  >  I  O  C  \  I  O  stations Figure 5.2.11  C  M  O  O  O  W  W  O  O  C  M  T  -  O  O  O  C  \  I  O  C  \  I  O  rf  Station Effect of l o g ( N 0 3 ) for Cluster 3 (monthly median/volume weighted mean (sorted by volume weighted mean), 1 9 8 3 - 1986)  O  Common Effect for Mean = 0.543 Common Effect for Median = 0.647  H  CM O  cn  '  -4  '  ne  \  S  EL E L LL  J .  L\  CM  s  9  modlan  msan  o  co co co co co co  CO CM O  «CD »-  CO r» CM  CM iO  CO CO CO ht- O  S  C —  8  O  C  l  O  O  C  O  O  C O C O T l O  C C  Q D  j  L  Q  O  C  C  O  C M  d '  C -  Q C  C O C O C O C O C O l O C  O M  C '  Q -  C C  O C O C O O f O C O  c o c M c o c M m c o i ^ c M c o c D i n c o ' < t i o c o T r T j C M O O O O O C M O O O O O O O O O O  stations  Figure 5.2.12  Boxplot for the resid. of log(N03) (83-86, monthly volume weighted mean, clust 1)  station  Figure 5.2.13(a)  Boxplot for the resid. of log(N03) (83-86, monthly median, clust 1)  station  Figure 5.2.13(b)  Boxplot for the resid. of log(N03) (83-86, monthly volume weighted mean, clust 2)  (TJ (5 CO W n J n j r o n j n s n j n j n J n J n j n j r o n J o j r o n j n j o j n j n j n J D _ . co to S O CO C CO CO tO CO N~— —K — — o o o ot-t-T-wtMocococoo^inwmtD, o o o o o o o o o o o o o o o o o o ^ O T - N C O O ) 0 ^ ( D C O ( J 1 0 ) T - O J O C D  station Figure  5.2.14(a)  injronironlfljfljrrjtT] (0 trj rO fd co CO O) T - CM CO i N C O i n N C O t D O W O ) ( M W W W W W N N W W O I N O I W O  Boxplot for the resid. of log(N03) (83-86, monthly median, clust 2)  i •  :  .  ! ; i  i  1  1  i !  ^Or-Ncocnot«D(OOio)i-(Mnfloorn(o[s  c o r o c o c d c o n j f f l c o c d c G  O » - r r W N n n C 0 W ( < ) T t U ) l 0 W « 0 S N S N S o o o o o o o o o o o o o o o o o o o o o  station Figure 5.2.14(b)  r o t n c i i r o r o c O R i c d t x j  W < \ t o J W C \ l < N O I N W N « O I N O i n  Boxplot for the resid. of log(N03) (83-86, monthly volume weighted mean, clust 3)  T O  -  N O  c O  g O  N  w O  c O  M O  w O  c  o O  c O  > c o O O  ^ O  T  f O  ^ O  ^ i O  n O  i  n O  i O  o ( O  station  Figure 5.2.15(a)  D O  ( -  D r  U -  5 ^  f  > C  c N  D J  i  o C  t N  ^  r J  ^ O  c J  o  u C  2 O  Boxplot for the resid. of log(N03) (83-86, monthly median, clust 3)  O) CO  o  rararamnjrarorararanjraairoioronjrararinjnjraranjranj O J O T - c > ] c o T r i n ^ w c o o ^ c o h * L o t D o o c O ' t i O L o ^ c o c \ i c o L O O o o o o o o o o o o o o o o o o o o o o o - t - - » - c \ i c \ J c \ j c o  station  Figure 5.2.15(b)  Summary of the Effects and Residuals from Median Polish of log(N03) (83-86,monthly volume weighted mean,clust 1)  1  i  i  1  1  -L  i  '  JL  Mon  Yr  Sit  I  :  .  i 1  1  1  I  i 1  r  I  I  •  I  MonYr  MonSit  YrSit  res  Figure 5.2.16(a)  CO  Summary of the Effects and Residuals from Median Polish of log(N03) (83-86,monthly median,clust 1) ,  —  _  ,  CJ  CO  cn  I CM  co  Mon  Yr  Sit  MonYr Figure 5.2.16(b)  MonSit  YrSit  res  CO  Summary of the Effects and Residuals from Median Polish of log(N03) (83-86,monthly volume weighted mean,clust 2) I  —  —  .  CM  J. _l_  I  co CO  CM  U  co  in Mon  Yr  Sit  MonYr Figure 5.2.17(a)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(N03) (83-86,monthly median,clust 2) CO  I  —  —  ,  CVI  ±  + I _L_  t  ca  in Mon  Yr  Sit  MonYr Figure 5.2.17(b)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(N03) (83-86,monthly volume weighted mean.clust 3)  Sit  MonYr Figure 5.2.18(a)  MonSit  Summary of the Effects and Residuals from Median Polish of log(N03) (83-86,monthly median,clust 3)  I  Mon  Yr  Sit  MonYr Figure 5.2.18(b)  MonSit  YrSit  res  Trend of log(N03) from 1983 to 1986 at the 81 Stations (calculated by monthly volume weighted mean)  O  0  no trend  -  down trend  +  up trend  Trend of log(N03) from 1983 to 1986 at the 81 Stations (calculated by monthly median)  Trend of log(N03) from 1983 to 1986 in the U S A (calculated by Kriging from monthly volume weighted mean)  -si  ro  Figure 5.2.20(a)  -  down trend  +  up trend  Trend of log(N03) from 1983 to 1986 in the USA (calculated by Kriging from monthly median)  CO  down trend  Figure 5.2.20(b)  up trend  TABLE RESULTS  5.1.1  THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRARTIONS OF NITRATE ( M o n t h l y V o l u m e W e i g h t e d Mean, '80 - '86)  site ID 004a 011a 017a 020a 021a 022a 023a 030a 031a 032a 034a 036a 038a 039a 040a 041a 049a 051a 052a 053a 055a 056a 058a 059a 064a 074a 075a 076a 168a 171a 173a  OF  Z  -1 .697 -1 .021 0 .152 -2 . 801 -0 .333 -2 .059 -2 .252 -1 .135 -1 .007 -1 .831 -1 . 970 -1 .684 -2 . 607 -1 .613 -0 . 604 -0 .711 -0 .890 -2 .269 -1 .279 0 . 449 -0 .732 -1 .025 0 . 406 -1 . 913 -0 .290 -4 .169 -0 .178 -3 .099 -1 .114 -0 .237 -3 .122  P- value  est'd slope  0 .090 0 .307 0 . 879 0 .005 0 .739 0 .039 0 .024 0 .256 0 .314 0 .067 0 .049 0 .092 0 .009 0 .107 0 .546 0 .477 0 .373 0 .023 0 .201 0 .653 0 .464 0 .305 0 . 685 0 .056 0 .772 0 .0 0 .859 0 .002 0 .265 0 .813 0 .002  -0 .003 -0 .002 0 .0 -0 .003 0 .0 -0 .003 -0 .003 -0 .003 -0 .001 -0 . 002 -0 .002 -0 .002 -0 .005 -0 .003 0 .0 -0 .001 -0 .001 -0 .004 -0 .002 0 .001 -0 .001 -0 .001 0 .0 -0 .005 0 .0 -0 .011 0 .0 -0 .004 -0 .002 0 .0 -0 .006  80 % L--bd -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0  174  .005 . 004 . 002 . 006 .003 . 006 . 005 . 006 .004 .004 .004 .004 .007 .006 .003 .003 .005 .006 .005 .002 .002 .004 .001 .010 .003 . 015 .002 .007 .005 .003 .009  u--bd  0 .0 0 .001 0 .002 -0 .002 0 .001 -0 .001 -0 .001 0 .0 0 .0 0 .0 0 .0 0 .0 -0 .002 0 .0 0 .001 0 .001 0 .0 -0 .001 0 .0 0 .004 0 .0 0 .0 0 .002 -0 .001 0 .001 -0 .008 0 .001 -0 .003 0 .0 0 .002 -0 .003  TABLE 5 . 1 . 2 RESULTS OF THE' MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRARTIONS OF NITRATE (Monthly Median, ' 8 0 - ' 8 6 ) site ID 004a 011a 017a 020a 021a 022a 023a 030a 031a 032a 034a 036a 038a 039a 040a 041a 049a 051a 052a 053a 055a 056a 058a 059a 064a 074a 075a 076a 168a 171a 173a  Z - 1 .. 954 - 0 ,. 7 4 6 0 ,. 5 3 8 - 2 .. 5 7 5 - 1 ,.383 - 3 ..236 - 1 .. 660 - 0 ..428 - 1 ..053 - 1 ..489 - 1 ..994 - 1 ..207 - 2 ,. 3 6 3 - 0 .. 9 2 5 - 0 .. 0 6 2 0 .. 3 8 1 - 1 ..240 - 2 ..248 - 1 ..146 - 0 .. 2 5 3 - 1 ..223 - 1 ..265 - 0 ..937 - 1 ..533 - 1 ..367 - 3 ..654 - 0 .. 5 9 2 - 3 ..064 - 2 .. 3 9 4 - 0 ..850 - 2 .. 0 8 0  p-value 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  . 051 .456 .591 .010 .167 .001 .097 .669 .293 .136 .046 .228 .018 .355 . 951 .703 .215 .025 .252 .800 .221 .206 .349 .125 .172 .0 .554 .002 .017 .395 .037  80% L'-bd U--bd  est'd slope -0 -0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 0 0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0  .003 .002 .001 .004 .002 .006 .003 .001 .002 .002 .003 .002 .004 .002 .0 .001 .002 .003 .003 .0 .002 .002 .001 .004 .002 .011 .001 .004 .004 .001 .004  -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0 -0  175  . 006 . 004 .001 .006 .004 .008 .006 .004 .004 .004 .005 .004 .007 .004 .002 .002 .006 .006 .005 .003 .003 .005 .002 .007 .004 .016 .003 .006 .007 .004 .007  -0 0 0 -0 0 -0 0 0 0 0 -0 0 -0 0 0 0 0 -0 0 0 0 0 0 0 0 -0 0 -0 -0 0 -0  . 001 . 001 . 003 .002 .0 .003 .0 .002 .0 .0 .001 .0 .001 .0 .002 .002 .0 .001 .0 .002 .0 .0 .0 .0 .0 .007 .001 .002 .002 .0 .001  TABLE 5.2.1 RESULTS OF THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRARTIONS OF NITRATE (Monthly Volume Weighted Mean, '83 - '86) site ID  z  P- value  est'd slope  L-•bd  004a 010a 011a 012a 017a 020a 021a 022a 023a 024a 025a 028a 029a 030a 031a 032a 033a 034a 035a 036a 037a 038a 039a 040a 041a 046a 047a 049a 051a 052a 053a 055a 056a 058a 059a 061a 063a 064a 065b 068a 070a 071a 073a  -2 .111 0 .514 -0 .382 0 .029 2 .757 -1 .256 -0 . 453 0 .258 -0 .222 -0 .880 -1 .093 0 .597 -1 .502 1 .202 1 . 119 -2 . 036 -0 .018 -1 .193 -2 .062 0 .284 -0 .767 -0 . 422 0 .507 1 . 840 1 .218 0 . 960 1 .730 0 . 018 0 .183 0 . 445 0 .841 -0 .364 1 .022 -0 .436 -2 .194 -0 .716 1 .229 0 .595 -1 .133 -1 .392 -0 .544 -1 .680 0 .734  0 .035 0 . 608 0 .702 0 . 977 0 .006 0 .209 0 .650 0 .797 0 .824 0 .379 0 .274 0 .550 0 .133 0 .230 0 .263 0 .042 0 .986 0 .233 0 .039 0 .776 0 .443 0 .673 0 .612 0 .066 0 .223 0 .337 0 .084 0 . 986 0 .854 0 .656 0 .400 0 .716 0 .307 0 .663 0 .028 0 .474 0 .219 0 .552 0 .257 0 .164 0 .586 0 .093 0 .463  -0.006 0.001 -0.001 0.0 0.009 -0.004 -0.001 0.001 -0.001 -0.003 -0.002 0.002 -0.007 0.007 0.005 -0.006 0.0 -0.003 -0.008 0.001 -0.005 -0.001 0.003 0.008 0.003 0.003 0.005 0.0 0.001 0.002 0.005 -0.001 0.003 -0.001 -0.016 -0.003 0.004 0.002 -0.003 -0.006 -0.002 -0.006 0.003  -0. 010 -0 .003 -0 .006 -0 .005 0. 004 -0 .008 -0. 005 -0. 004 -0. 005 -0 .009 -0 .005 -0. 002 -0. 011 0. 0 -0. 001 -0 .011 -0. 004 -0. 007 -0. 015 -0. 003 -0. 014 -0. 004 -0 .005 0 .003 0. 0 -0. 001 0 .001 -0 .008, -0. 005 -0. 004 -0. 003 -0. 004 -0. 001 -0. 004 -0. 026 -0. 011 0. 0 -0. 002 -0. 007 -0. 014 -0. 008 -0. 013 -0. 002  1 76  80%  U--bd  -0 .002 0 .006 0 .004 0 .005 0 .014 0 .0 0 .002 0 .004 0 .004 0 .002 0 .0 0 .007 -0 .001 0 .014 0 .009 -0 .002 0 .003 0 .0 -0 .004 0 .005 0 .004 0 .002 0 .010 0 .012 0 . 007 0 .007 0 .008 0 .008 0 .006 0 .007 0 .013 0 .002 0 .008 0 .003 -0 .006 0 .002 0 .008 0 .006 0 .0 -0 .001 0 .004 -0 .001 0 .008  TABLE 5.2.1  (continued)  RESULTS OF THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRARTIONS OF NITRATE . (Monthly Volume Weighted Mean, '83 - '86) site ID  Z  P- value  est'd slope  L--bd  074a 075a 076a 077a 078a 160a 161a 163a 164a 166a 168a 171a 172a 173a 249a 251a 252a 253a 254a 255a 257a 258a 268a 271a 272a 273a 275a 277a 278a 279a 280a 281a 282a 283a 285a 349a 350a 354a  -3.,799 3..511 -0 . .770 -0..257 -1..204 -2 . 974 0 ,116 . 1.. 941 1.,013 0 .0 0. 975 1. 902 0 .914 -3. 013 1. 096 1. 702 -1. 698 -0. 761 -1. 013 -0. 667 1. 598 -0 .502 0. 293 -0 .479 -1. 253 -1. 902 -0. 107 1. 816 -3. 329 -2 .32 9 -1. 351 -0. 431 1. 129 -1. 247 1. 411 1. 506 1. 879 -4. 106  0 .0 0 .0 0 .441 0 .797 0 .229 0 .003 0 .908 0 .052 0 .311 1 .000 0 .329 0 . 057 0 .361 0 .003 0 .273 0 .089 0 .090 0 . 447 0 .311 0 .505 0 .110 0 . 616 0 .769 0 . 632 0 .210 0 . 057 0 . 915 0 .069 0 .001 0 .020 0 .177 0 .666 0 .259 0 .212 0 .158 0 .132 0 .060 0 .0  -0 .021 0 . 009 -0 .003 -0 .001 -0 .005 -0 .014 0 .001 0 .005 0 .006 0 .0 0 .003 0 .007 0 .0 -0 .009 0 .005 0 .011 -0 .007 -0 .003 -0 .005 -0 . 003 0 .008 -0 .002 0 .002 -0 .003 -0 .004 -0 .005 0 .0 0 .012 -0 .014 -0 .011 -0 .005 -0 .003 0 .004 -0 .003 0 .008 0 .004 0 .011 -0 .028  -0 .030 0 . 006 -0 . 006 -0 .007 -0 .012 -0 . 019 -0 .005 0 .001 -0 .002 -0 .004 -0 .001 0 .002 0 .0 -0 .014 -0 .001 0 .003 -0 .011 -0 .009 -0 .012 -0 .011 0 . 001 -0 . 006 -0 .003 -0 .013 -0 .009 -0 .010 -0..006 0 .004 -0 .022 -0 .018 -0,.011 -0 .013 -0 .001 -0 .005 0 .0 0 .001 0 .003 -0 .040  177  80 %  U--bd  -0 .014 0 .012 0 .002 0 . 004 0 .0 -0 .008 0 . 005 0 . 010 0 .014 0 .005 0 .008 0 .011 0 .005 -0 .006 0 .012 0 .020 -0 .002 0 .002 0 .001 0 .003 0 .017 0 .004 0 .006 0 .003 0 .0 -0 .002 0 .005 0 .018 -0 .009 -0 .004 -0 .001 0 .006 0 .009 0 .0 0 .016 0 .006 0 .019 -0 .018  RESULTS OF THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRARTIONS OF NITRATE (Monthly Median, ' 8 3 - ' 8 6 )  z  s i t e  ID 004a 010a 011a 012a 017a 020a 021a 022a 023a 024a 025a 028a 029a 030a 031a 032a 033a 034a 035a 036a 037a 038a 039a 040a 041a 046a 047a 049a 051a 052a 053a 055a 056a 058a 059a 061a 063a 064a 065b 068a 070a 071a 073a  -1 0 0 0 1 -0 -0 -1 0 -2 -0 0 -0 1 0 -2 0 -1 -1 0 -0 -0 1 1 2 0 1 0 0 0 1 -0 1 0 -1 -1 1 0 -0 -0 -0 -1 0  .577 .138 .124 .127 .781 .935 .711 .467 . 667 .391 . 667 .020 . 649 .431 .248 . 054 . 071 . 827 . 874 .258 .388 .165 .786 .084 .0 62 .560 .760 .471 . 954 .767 .350 .009 .022 .827 .992 .243 .100 .782 .734 .511 .680 .147 .724  P~ v a l u e 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  .115 .891 . 901 .899 .075 .350 .477 .143 .505 . 017 .505 .984 .516 .153 .804 .040 . 943 .068 .061 .797 . 698 .869 .074 .278 .039 .576 .078 .638 .340 .443 .177 .993 .307 .408 .046 .214 .271 .434 .463 .609 .496 .252 .469  e s t ' d slope -0 0 0 0 0 -0 -0 -0 0 -0 -0 0 -0 0 0 -0 0 -0 -0 0 -0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0 -0 0 0 -0 -0 -0 -0 0  .006 .0 .001 .001 .006 .003 .003 .005 .003 .009 .002 .0 .003 .007 .001 .008 .0 .008 .010 .001 .002 .0 .009 .004 .006 .002 .006 .003 .003 .004 .007 .0 .004 .003 .011 .009 .003 .002 .003 .002 .002 .006 .003  80  L-- b d -0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0 -0 -0 0 -0 0 -0 0 -0 -0 -0 0 -0 -0 -0 -0 -0 0 -0 -0 -0 -0 -0 -0  178  .012 .003 .004 .006 .001 .007 .007 .010 .002 .016 .005 .005 .010 .001 .004 .014 .004 .014 .016 .003 .010 .004 .003 .001 .002 .002 .001 .005 .001 .003 .0 .003 .001 .002 .022 .016 .0 .001 .008 .010 .008 .014 .003  % U'- b d -0 0 0 0 0 0 0 -0 0 -0 0 0 0 0 0 -0 0 -0 -0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 -0 0 0 0 0 0 0 0 0  . 001 . 003 . 005 .006 .012 .001 .002 .001 .007 .004 .002 .006 .003 .014 .005 .003 .003 .003 .004 .004 .006 .003 .017 .009 . 010 .006 .009 .011 .007 .011 .014 .004 .010 .007 .003 .0 .007 .008 .002 .004 .004 .001 .008  TABLE 5.2.2 RESULTS  site  ID  074a 075a 076a 077a 078a 160a 161a 163a 164a 166a 168a 171a 172a 173a 249a 251a 252a 253a 254a 255a 257a 258a 268a 271a 272a 273a 275a 277a 278a 279a 280a 281a 282a 283a 285a 349a 350a 354a  (continued)  OF THE MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRARTIONS OF NITRATE ( M o n t h l y M e d i a n , '83 - '8 6)  z -2 .039 2 .711 -1 .495 -0 .330 -0 .489 -3 . 933 0 .338 0 . 407 1 .165 -0 . 917 • 0 . 104 1 . 627 3 .227 -1 .964 1 .252 1 .772 -1 .733 -0 .715 -0 .218 -0 .844 0 .850 -0 .739 -0 .373 -0 .108 -0 .080 -2 .195 0 . 142 2 . 623 -3 .299 -1 .964 -1 .12 9 -0 .138 0 .702 -0 .972 1 .155 1 .278 1 .106 -3 .486  p - value  0 .041 0 .007 0 .135 0 .741 0 . 625 0 .0 0 .736 0 . 684 0 .244 0 .359 0 .917 0 . 104 0 .001 0 .049 0 .210 0 .076 0 .083 0 .474 0 .828 0 .398 0 .396 0 .460 0 .709 0 . 914 0 . 936 0 .028 0 .887 0 .009 0 .001 0 .050 0 .259 0 .891 0 .483 0 .331 0 .248 0 .201 0 .269 0 .0  est'd slope  L--bd  -0 .013 0 .010 -0 .004 -0 .001 -0 .003 -0 .018 0 .001 0 . 002 0 .008 -0 .003 0 .0 0 . 006 0 .010 -0 .009 0 .003 0 .013 -0 .006 -0 .002 -0 .002 -0 .005 0 .005 -0 .003 -0 .002 0 .0 0 .0 -0 .008 0 .001 0 .019 -0 .015 -0 .010 -0 .005 -0 .001 0 .003 -0 .002 0 .007 0 .004 0 .007 -0 .018  80%  -0 .022 0 .005 -0 .008 -0 . 004 -0 .011 -0 .025 -0 .003 -0 .005 0 .0 -0 .008 -0 .006 0 .002 0 .006 -0 .015 0 .0 0 .005 -0 .011 -0 .007 -0 .010 -0 .013 -0 .002 -0 .008 -0 .007 -0 .009 -0 .005 -0 .013 -0 .005 0 .011 -0 .023 -0 .017 -0 .012 -0 .008 -0 .003 -0 .006 -0 .001 0 .0 0 .0 -0 .026  179  u--bd  -0 .004 0 . 016 -0 . 001 0 .002 0 .004 -0 .012 0 .004 0 .006 0 .015 •0.001 0 .006 0 .011 0 .013 -0 .003 0 .011 0 .021 -0 .002 0 .002 0 .004 0 .003 0 .011 0 .002 0 .003 0 .006 0 .003 -0 .003 0 .006 0 .027 -0 .009 -0 .004 0 .001 0 .011 0 .009 0 .001 0 . 011 0 .008 0 .015 -0 .012  Chapter 6  A N A L Y S E S AND CONCLUSIONS F O R H Y D R O G E N ION  Monthly precipitation volume weighted mean and monthly median are referred to as "weighted mean" and "median", respectively, below.  6.1  Results for the Historical Data "Historical data" refers to the data collected from January, 1980 to December, 1986. For  H  +  (calculated from pH), there are 32 sites with 5 or fewer missing observations during the  period. The locations of these 32 sites are plotted in Figure 6.1.1. From this figure we can see that more sites are located in the East than the West. The results of this section are based on the data obtained from these 32 sites.  6.1.1  Clustering and Transformation  Figure 6.1.2 shows the hierarchical cluster structure of the weighted means. The sites are partitioned quite naturally into three clusters.  The locations of the sites labelled by  clusters are shown in Figure 6.1.3. The pattern is obvious: clusters 3 and 1 are located in the subregion of highest industrial concentration and the surrounding regions; cluster 2 is spread over the rest of the United States.  This indicates that the pattern of wet acid  deposition in the industrial areas is different from that of the other areas. Note that cluster 2 can be split into two subgroups according to Figure 6.1.2, one subgroup is located along the East Coast and the other spread over the West and the middle of the United State. This suggests again that the geographical location and the manner of wet acid deposition are closely related.  180  The histograms of the original data and the transformed data under the three kinds of transformations described in Section 3.1, are shown in Figure 6.1.4. histograms appears symmetric.  None of the four  The ones for 1/4 power transformation and for log  transformation seem better than the other two.  Since the log transformation of H  +  is  equivalent to the p H value, which gives us a familiar measure of acidity, the log transformation is adopted in this case and the remainder of our analysis of the hydrogen ion data is based on the transformed data. The histograms of the log transformed data for each cluster are depicted in Figure 6.1.5. It seems that the histogram for cluster 3 fits the normal curve well and that for cluster 1 it is approximately symmetric. The one for cluster 2 is left skewed.  6.1.2  T r e n d , Seasonality a n d Spatial Patterns  The analysis described below is done for both log transformed monthly volume weighted mean and log transformed monthly median H  +  data. The analysis is applied separately to  each cluster obtained from clustering the volume weighted mean data. Three-way median polishes are applied to both weighted means and medians in each cluster, in the manner described in Section 3.2.  Figures 6.1.6 and 6.1.7 show the plots of  the yearly effects of the three clusters for weighted mean and median, respectively.  In  general they both show a V-shaped pattern where the one for weighted means is more obvious.  In Figure 6.1.6, clusters 1 and 3 show down-trends before 1983 and up-trends  after 1983.  Cluster 2 behaved slightly differently, namely, the suggested change point is  1984 instead of 1983. Figure 6.1.7 shows a pattern like the one in Figure 6.1.6 except that the estimated effect of 1982 for cluster 2 is different.  The reason for this needs further  study. Generally speaking, Figures 6.1.6 and 6.1.7 suggest that the concentration of H in +  wet deposition decreased before 1983 and increased after 1983 for the most industrialized  1 81  subregion of the United States. They also suggest that for the other regions of the United States, there is a similar V-shaped trend. Figure 6.1.8 and Figure 6.1.9 show the monthly effects for the weighted means and medians, respectively.  They have almost the same pattern. The patterns for all three  clusters are similar as well. The effects are high from June to September and low from November to April. The transitions from month to month are quite smooth and there is a peak in August. The fact that the seasonal patterns are very similar for all three clusters suggests a reassuring spatial stability in this seasonal pattern.  Figures 6.1.10 to 6.1.12 display the site effects of log(H ) by cluster; the weighted +  means and medians are plotted together and sorted by weighted means within each cluster. In Figure 6.1.10, the stations located close to the center of the industrial area tend to have large positive effects and the stations located far from the center of the industrial area tend to have large negative effects.  In Figure 6.1.11, there is an interesting pattern: all the  stations with positive effects are located in the Eastern United States and all the stations with negative effects are located in the West or the middle of the United States.  Figures 6.1.13 to 6.1.15 are the boxplots of the residuals of the three-way median polish of the log(H ) data where (a) is for weighted means and (b) is for medians and each box +  represents one site.  In general, the variations of the residuals in each cluster are of the  same order. The differences between the boxplots for the means and for the medians are even smaller. The variation of the residuals is relatively small for cluster 3 and relatively large for cluster 2. In Figures 6.1.13 ((a) and (b)), we observe a few extreme outiiers from Station 021a, 032a and 051a. In Figures 6.1.14 ((a), (b)), extreme outliers are observed at Stations 011a, 017a and 053a. In Figures 6.1.15 ((a), (b)), the boxes for Stations 041a have extreme outliers.  Note that some other stations have outliers as well.  182  A l l the stations  with outliers should be carefully checked and further studied. But no analyses for specific site are included in this paper. Figures 6.1.16 to 6.1.18 summarize the median polish for the three clusters, respectively where (a) is for the weighted means and (b) is for the medians.  In each figure, boxes  represent the main effects, interactions and residuals respectively. We regard interactions as non-negligible if the sizes of the boxes for the interactions are comparable with those for main effects.  In these figures, the boxes for yearly effects are smaller indicating that  compared with seasonal variation and site variation, the yearly changes are smaller.  The  boxes for the interactions of month-by-year and month-by-site have the sizes comparable to those of the main effects, which suggests that they have to be taken into account. For cluster 2, the box for site effect is very large. This may be caused by the fact that cluster 2 is the largest cluster spreading over the United States, and hence the variation of site effects is large (see Figures 6.1.17 (a) and (b)).  Figures 6.1.18 (a) and (b) show the  summaries for cluster 3. The fact that the boxes for site effects and their interactions are very small suggests that the four sites behaved very similarly.  6.1.3  The Results of Trend Testing, Slope Estimation and Kriging  Mann-Kendall Test and Sen's nonparametric slope estimation procedure are applied to the deseasonalized data for each site in the manner described in Section 3.3. Kendall  A Mann-  Test statistic with the corresponding p-value and Sen's (1968b) nonparametric  slope estimate with its 80% confidence interval are obtained for each site for both weighted means and medians. These are shown in Tables 6.1.1 and 6.1.2 respectively.  In Table  6.1.1, 11 out of the 32 sites have significant trends at the p=0.2 level. In Table 6.1.2, 10 out of the 32 sites have significant trends at the p=0.2 level.  The results of the nonparametric slope estimation are plotted in Figures 6.1.19 (a) and (b) with symbols defined in Section 3.3, for the weighted means and medians respectively. 183  If the lower 80% confidence bound of the slope estimate is greater than 0 at any site a "+" is plotted; if the upper 80% confidence bound of the slope estimate is less than 0 a "-" is plotted. Otherwise, a "0" is plotted . "+", "-" and "0" represent respectively, an up-trend, a down-trend and no-trend. The patterns of these two figures are very similar. In Figure 6.1.19 (a), 7 of the sites show down-trends and they are mainly located in the Eastern United States.  The two located along the West Coast show up-trend. Other sites show  no-trend. Figure 6.1.19 (b) shows a slightly different pattern in which two sites located in the South show up-trend. The main result is that during the period of 1980 - 1986, most of the sites have no-trend. However, keeping Figures 6.1.6 and 6.1.7 in mind, the results of trend tests and slope estimation may be misleading in case of the presences of a V-shaped trend. The results of Kriging the slope estimates are displayed in Figures 6.1.20(a) and (b). As before, Figure 6.1.20(a) is for the weighted means and (b) is for medians. A "+" means an up-trend is estimated at the corresponding point and a "-" means a down-trend, estimated at the point.  Since the estimated variograms used for these two plots are both  constant, Figures 6.1.20 (a) and (b) are actually produced by moving averages.  Hence  these two figures are relatively rough and should be read in conjunction with Figure 6.1.19. At any particular point where the observations were not made, interpolation may not be reliable.  This is because (a) estimation is based on the estimated slopes, not on the  observations directly and (b) the spatial resolution of the measurements is low and therefore, the correlations between sites are small.  6.2  Results for the Recent Data "Recent data" refers to the data collected from January, 1983 to December, 1986. For  H , there are 86 sites with 5 or fewer missing observations during the period. +  1 84  The  locations of these 86 sites are plotted in Figure 6.2.1.  From this figure we can see that  more sites are located in the Northeast than the rest of the areas.  The results of this  section are based on the data observed at these 86 sites.  6.2.1 Clustering and Transformation Figure 6.2.2  shows the hierarchical cluster structure of the weighted means. The sites  are partitioned naturally into three clusters. The locations of the sites labelled by cluster are shown in Figure 6.2.3.  It gives a pattern similar to the one in Figure 6.1.3: cluster 2 is  located in the subregion of highest industrial concentration; cluster 1 is located around that region; and cluster 3 is located throughout the rest of the United States, corresponding to cluster 3, 1 and 2 in Figure 6.3.1 respectively.  This gives us the same indications that we  mentioned in Section 6.1.1, namely, that the wet acid deposition in the industrial areas has a different manner from that of the other area. The histograms of the original data and the transformed data under the three kinds of transformation described in Section 3.1, are shown in Figure 6.2.4.  It appears that the  histogram of the 1/4 power transformed data fits the normal curve slightly better than that of the log transformed data. For the reason given in Section 6.1.1, also for the reason that the log transformation is used in the case of "historical data", the log transformation is adopted for our analysis so that comparisons can be easily made.  The histograms of the log  transformed data for each cluster are depicted in Figure 6.2.5. It appears that the histogram for cluster 2 fits the normal curve well. The histogram of cluster 1 fit the normal curve too but it has a light long left tail. The histogram of cluster 3 is left skewed.  6.2.2  Trend, Seasonality and Spatial Patterns  185  The analysis described below is applied to both log transformed monthly volume weighted means and log transformed monthly medians. The analysis is applied separately to each cluster obtained from clustering the volume weighted mean data. Three-way median polishes are applied to both weighted means and medians in each cluster. Figures 6.2.6 and 6.2.7 show the plots of the yearly effects of the three clusters for weighted mean and median respectively. They show similar patterns. In general, all three clusters show up-trends. This conclusion, obtained from 86 stations instead of 32, is consistent with what we got from Figures 6.1.7 and 6.1.8. That is, during the period of 1983 to 1986, the concentration of H  +  in wet deposition increased for the most industrialized  subregion of the United States, and for most of the other regions of the United States as well. Figures 6.2.8 and 6.2.9 show the monthly effects for the weighted means and medians, respectively.  Although more than half of the data used in the present analyses were not  included in the historical data, the patterns are quite close to those for the latter displayed in Figures 6.1.8 and 6.1.9: high from June to September; low from November to April with a peak in August.  This convinces us again of spatial stability in the seasonal pattern.  Besides, it seems more clearly that there is a gap in April for cluster 3.  Figures 6.2.10 to 6.2.12 show the site effects of log(H ) by cluster; the weighted means +  and medians are plotted together and sorted by weighted means within each cluster. In Figure 6.2.10, the difference between the effects of the weighted means and the medians at Station 021a is large. To see how this occurred we need to study the original data further. For cluster 2, the stations located in the center of the industrial region tend to have larger effects.  Figures 6.2.13 to 6.2.15 are the boxplots of the residuals of the three-way median polish of  the log(H ) data where (a) is for weighted means and (b) is for medians and each box +  186  represents one site.  In general, the variations of the residuals in each cluster are of the  same order and are comparable to those in Figures 6.1.13 to 6.1.15.  The differences  between the boxplots for the means and for the medians are even smaller. The boxes for Station 035a and 038a in both Figures 6.1.15 (a) and (b) are larger indicating that the data from these two station do not fit the model well. Extreme outliers are found in Stations 021a, 023a, 025a,039a, 051a, 053a and 161a in Figures 6.2.13 (a) or (b), 058a in Figures 6.2.14 (b) and 015a, 024a, 029a, 068a, 071a, 077a, 225a and 339a in Figures 6.2.15 (a) or (b).  Among these stations, 021a, and 053a have extreme outliers observed in Figures  6.1.13 to 6.1.15 and those outliers may be the same ones.  A l l these sites and the other  stations with outliers need further study. Figures 6.2.16 to 6.2.18 summarize the result of the median polish for the three clusters respectively where (a) is for weighted means and (b) is for medians. In each figure, boxes represent the main effects, interaction and residuals. These figures are similar to those in Figures 6.1.16, 6.1.18 and 6.1.17 respectively. The interpretation for them can be applied here correspondingly. Generally speaking, cluster 2 fit the model best.  6.2.3  T h e Results of T r e n d Testing, Slope Estimation a n d K r i g i n g  The Mann-Kendall Test and Sen's slope estimation procedure which were applied to "historical data" are used for "recent data" as well. A Mann-Kendall test statistic with the corresponding p-value and Sen's (1968b) nonparametric slope estimate with its 80% confidence interval are obtained for each site for both weighted means and medians. These are shown in Tables 6.2.1 and 6.2.2 respectively.  In Table 6.2.1, 39 out of the 86  S l t e s  h  a  v  e  significant trends at the p=0.2 level. In Table 6.2.2, 40 out of the 86 sites have significant trends at the p=0.2 level.  Figures 6.2.19 (a) and (b) are obtained by the same method used to produce Figures 6.1.19, for weighted means and medians respectively. Again, "+", "-" and "0" represent an  187  up-trend, a down-trend and no-trend, respectively.  In Figure 6.2.19 (a), 10 sites show  down-trends and are located in the middle east of the United States.  Slightly more than  half of the sites show no-trend and the rest of the sites show up-trend. Those sites are located throughout the United States. The pattern of Figure 6.2.19 (b) is quite similar to Figure 6.2.19 (a) except that there are a few more "up-trend" stations and fewer "downtrend" stations. Comparing Figures 6.2.19 ((a) and (b)) with Figures 6.1.19 ((a) and (b)), we can see that a lot more stations with "+" appear in the former. Some of the stations with "-" in Figure 6.1.19 have "+" in Figures 6.2.19. This implies that some stations do have a V-shaped trend during the period of 1980 - 1986.  The results of Kriging the slope estimates are displayed in Figures 6.2.20(a) and (b) where (a) is for the weighted means and (b) is for the medians. As before, a "+" means an up-trend and a "-" means a down-trend estimated at the corresponding point, respectively. Again, these two plots are similar except that both the "+" area and "-" areas in Figure 6.2.20 (b) are smaller than those in Figure 6.2.20 (a).  Both figures show that there is an  up-trend for the East Coast area, a down-trend in certain areas of the middle of the United States and no-trend in the rest of the areas.  For the reason given in Section 6.1.3, these  two figures can only be used as indicators of the spatial distribution of the trends as the interpolation error for a specific point could be large.  188  The Locations Of The 32 Monitoring Stations From 1980 To 1986 (For Hydrogen ion)  Clustering of H+ monthly volume weighted mean based on sqrt(MSE) 1-989-4986-1  r  s 8  8  8  &  Figure 6.1.2  3  C l u s t e r s of H+ monthly v o l u m e w e i g h t e d m e a n b a s e d on sqrt(MSE)  1980- 1986 (k=3)  Figure 6.1.3  Histograms of Transformed H+ (Volume Weighted Mean, 80-86) Histgram ol H+ (volume weighted mean, 80 - 86)  50  100  150  200  Histgram ol log(H+) (volume weighted mean, 80 - 86)  250  300  CD  ro Histgram of sqrt(H+) (volume weighted mean, 80 - 86)  Histgram of (H+) (1/4) (volume weighted mean, 80 - 86) A  S 8 n  s . CM  S  Figure 6.1.4  i  1  1  1  1  1  1  1  0.5  1.0  1.5  20  2.5  3.0  3.5  4.0  1 «.5  Histograms of log(H+) by Clusters (Volume Weighted Mean, 80-86) Histogram lor Cluster 1 ol log(H+) (vwm, 80 - 86) based on H+  Histogram lor Cluster 2 ot log(H+) (vwtn, 80 - 86) based on H+  8  C D C O  Histogram for Cluster 3 of log(H+) (vwm, 80 - 86) based on H+  Figure 6.1.5  .  Yearly Effect of log(H+) for 3 Clusters (monthly volume weighted mean, 1980 - 1986)  \! 3  1980  1981  1982  1983  Figure  6.1.6  1 94  1984  1985  1986  Yearly Effect of log(H+) for 3 Clusters (monthly median, 1980 - 1986)  1980  1981  1982  1983  Figure  6.1.7  1 95  1984  1985  1986  Monthly Effect of log(H+) for 3 Clusters (monthly volume weighted mean, 1980 - 1986) 2  , Jan 1  J Mar  1  May  1  1  Jul  Sep  Figure 6.1.8  1 96  1  Nov  2  Jan  Monthly Effect of log(H+) for 3 Clusters (monthly median, 1980 - 1986)  1  Jan  1  Mar  1  May  1  Figure  Jul 6.1.9  1 97  1  Sep  1  Nov  2  Jan  Station Effect of log(H+) for Cluster 1 (monthly median/volume weighted mean (sorted by volume weighted mean), 1980 - 1986) Common Effect for Mean = Common Effect for Median =  3.734 3.709  T  / / / / /  / /  /  /  \>  / /  \  / \  / 7  / / / / /  \  /  \  /  \  /  \ \  / \  \  —|  \ \ \ \  / \ \  /  \  \  \ \  \  median  \  mean  \ \  V Si  (0 to  to oo to  stalions  Figure 6.1.10  to  to m  to  nj m in  \  / \ / \ / \ / \ / \  / \ /  N  Station Effect of log(H+) for Cluster 2 (monthly median/volume weighted mean (sorted by volume weighted mean), 1980 -1 Common Effect for Mean = Common Effect for Median =  2.350 2.393  \  V\N P N T j s r  /  /  /  a  / /  /  /  / / / /  median  / / / / CO C CO O  nl cn to  nj co  ra  stations Figure 6.1.11  moan  Station Effect of log(H+) for Cluster 3 (monthly median/volume weighted mean (sorted by volume weighted mean), 1980 - 1986) Common Effect for Mean = Common Effect for Median =  4.085 4.177  stations  Figure 6.1.12  Boxplot for the resid. of log(H+) (80-86, monthly volume weighted mean, clust 1)  o  8  CO CM CM  CO CM CO  CO  o station  Figure 6.1.13(a)  CO  Boxplot for the resid. of log(H+) (80-86, monthly median, clust 1)  slation  Figure 6.1.13(b)  Boxplot for the resid. of log(H+) (80-86, monthly volume weighted mean, clust 2)  T  to •>* o o  co o o  CO o  CO IS o  CO o cn o  CO "3CO o  CO CD CO O  CO oo CO o  CO  01 o  station Figure 6.1.14(a)  CO CM in o  CO CO in o  co  01 m o  to o  «  I  CO CD  to CO  o  Boxplot for the resid. of log(H+) (80-86, monthly median, clust 2)  ro o 4^  ro  ra  CO to  o station  Figure 6.1.14(b)  Boxplot for the resid. of log(H+) (80-86, monthly volume weighted mean, clust 3)  station  Figure 6.1.15(a)  Boxplot for the resid. of log(H+) (80-86, monthly median, clust 3)  i  ra in  station  Figure 6.1.15(b)  Summary of the Effects and Residuals from Median Polish of log(H+) (80-86,monthly volume weighted mean,clust 1)  Sit  MonYr Figure  6.1.16(a)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(H+) (80-86,monthly median,clust 1)  Mon  Sit  MonYr Figure 6.1.16(b)  MonSit  YrSit  res  CO  Summary of the Effects and Residuals from Median Polish of log(H+) (80-86,monthly volume weighted mean,clust 2)  CM  ro o CD  CM  CO  Sit  MonYr  Figure 6.1.17(a)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(H+) (80-86,monthly median,clust 2)  ro o  o  h  OJ  Sit  MonYr Figure 6.1.17(b)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(H+) (80-86,monthly volume weighted mean,clust 3)  T  _T_ I  ±  I  I  !  ±  Mon  Yr  Sit  MonYr  Figure 6.1.18(a)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(H+) (80-86,monthly median.clust 3)  •  Mon  Yr  Sit  MonYr Figure 6.1.18(b)  MonSit  YrSit  Trend of log(H+) from 1980 to 1986 at the 32 Stations (calculated by monthly volume weighted mean)  0  no trend  -  down trend  +  up trend  Trend of log(H+) from 1980 to 1986 at the 32 Stations (calculated by monthly median)  215  Trend of log(H+) from 1980 to 1986 in the USA (calculated by Kriging from monthly median)  Figure 6.1.20(b)  -  down trend  +  up trend  The Locations Of The 86 Monitoring Stations From 1983 To 1986 (For Hydrogen ion)  Figure 6.2.1  Clustering of H+ monthly volume weighted mean based on sqrt(MSE)  r  •  1983 - 1  Figure 6.2.2  Clusters of H+ monthly volume weighted mean based on sqrt(MSE) 1983- 1986 (k=3)  Histograms of Transformed H+ (Volume Weighted Mean, 83-86) Histgram of H+ (volume weighted mean, 83 - 86)  0  50  1 00  150  200  Histgram of log(H+) (volume weighted mean, 83 - 86)  250  300  6  -4  2  0  2  4  6  ro ro o Histgram of sqrt(H+) (volume weighted mean, 83 - 86)  8  -  Figure 6.2.4  Histgram of (H+) (1/4) (volume weighted mean, 83 - 86) A  9  Histograms of log(H+) by Clusters (Volume Weighted Mean, 83-86) Histogram for Cluster 2 of log(H+) (vwm, 83 - 86) based on H+  Histogram lor Cluster 1 of log(H+) (vwm, 83 - 86) based on H+  -i  1  —  -  i  Histogram for Cluster 3 of log(H+) (vwm, 83 - 86) based on H+  Figure 6.2.5  i  i  2.0  2.5  -i  r-  3.0  3.5  4.0  r" 4  5  LJZL 5.0  I  1  5.5  8.0  Yearly Effect of log(H+) for 3 Clusters (monthly volume weighted mean, 1983 - 1986)  1983  1984  1985 Figure 6.2.6  222  1986  Yearly Effect of log(H+) for 3 Clusters (monthly median, 1983 - 1986)  1983  1984  1985  Figure 6.2.7  223  1986  Monthly Effect of log(H+) for 3 Clusters (monthly volume weighted mean, 1983 - 1986) 2 .1.  \  i  \  \  \  \  \ /  /  i  * i  3  1  Jan  1  Mar  1  1  May  Jul Figure  6.2.8  224  1  1  Sep  Nov  2 Jan  Monthly Effect of log(H+) for 3 Clusters (monthly median, 1983 - 1986) ii  '2' i  /• /•  i  r •»  1  Mar  1  1  May  1  Jul Figure 6.2.9  225  1  Sep  Nov  Station Effect of log(H+) for Cluster 1 (monthly median/volume weighted mean (sorted by volume weighted mean), 1983 -1 Common Effect for Mean = Common Effect for Median =  3.503 3.479  /  /  -  \ /  \ /  V  \ /  /  s /  s  1  ro ro  CD  \  median  mean  L\  to  d  ro CO in  o  ro co  ro cn  o  o  ro ro  If)  CM  CO  i-  SI o  ro ro ro ro  CO CM O  i—  CO O  r—  CO T-  Stations  Figure 6.2.10  CO CO r-  ro ro ro ro tnooro  CM CO O  T—  CM O  o CO  IT)  CM r~ CM  CM  Station Effect of log(H+) for Cluster 2 (monthly median/volume weighted mean (sorted by volume weighted mean), 1983 - 1986) Common Effect for Mean = Common Effect for Median =  4.011 4.026  / /  T  /  /  / / /  /  /  \ / \  /  /  \ \ \ \  \  / \  \  / \  \  / \  \  \  \  \  \  \  \  \  7  /  /  /  y  /  /  /  / \ / /  / /  \  \ \ \ \  \  /  mfldian  \ (0  stations  Figure 6.2.11  mtan  Station Effect of log(H+) for Cluster 3 (monthly median/volume weighted mean (sorted by volume weighted mean), 1983 Common Effect for Mean = Common Effect for Median =  2.160 1.918  ro ro co  median  stations Figure 6.2.12  mean  1986)  Boxplot for the resid. of log(H+) (83-86, monthly volume weighted mean, clust 1)  l  I  I  T  _L  T  f  L I *  1  I  ro ro  I  i  i  1  *  I  i  +  X  CO  t a c G a i t t i t t i B i t B t a t Q a s a i G i t a a i e i s c a c a t o a i a i a i t t i Q t n  O  C O  r  -  C  M C M O O  A  C O  i  \  c  O  i  o  C O  u  \ O  o J  o  O  C  D  M O  T  C O  -  M  c  O  C  M  O O  c  M O  o  M  c  O  C  n  O O  c  ^ -  ^  r r  T  U  -  -  7  c  T  U  v  l -  j  i  c  »  n  o -  station Figure  6.2.13(a)  r  c  ^ C  O  C  T  a  D  -  c  C  ^  D \  '  C J  c  o  D C  ^  c \  L  n  i  O  ^  l C  -  O  r  M  f  ^  ^ C  c  M  C M  O  i  C  L  o  O 0  o  Boxplot for the resid. of log(H+) (83-86, monthly median, clust 1)  CO  8  ra  CO CJ o  03  in CM o  nl 00 CM  O  nl CO o  ro  CM CO O  ra  CO CO o  ro  cn CO o  nl cn o  nl  in o  ro  to ro m t^  CM  CO  CO  o  o  o  in  station  Figure 6.2.13(b)  ra CO  ro CO  ro  00 CO  ro  *r  CM  ro  in CM  to in CM  to  CM  CM  ra  in 00 CM  ra  o  in CO  Boxplot for the resid. of log(H+) (83-86, monthly volume weighted mean, clust 2)  ro  CO  + X  station Figure  6.2.14(a)  Boxplot for the resid. of log(H+) (83-86, monthly median, clust 2)  station Figure  6.2.14(b)  (O  co  Boxplot for the resid. of log(H+) (83-86, monthly volume weighted mean, clust 3)  _  I  !  caconjcororocorotdconinjrarooanjcacaronjffl ^i^o*-c\iincor^Tfcno-^in<or^rocnT-c>jmoi-^<or^o30co<oc\imc^ o o T - ^ r - t - r - ^ - w w n o n n c ^ n ^ n l O ^ O ( 0 ^ ^ ^ ^ ^ ^ ( D a ^ o ^ ^ l n l n l n l f l l n f f l ^ ^ ^ ^ ^ c o l o c o c o n * l n O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  O  T  -  T  -  T  -  ^  ^  C  \  I  C  M  C  \  J  C  V  1  C  station  Figure 6.2.15(a)  M  C  \  J  C  M  C  \  J  C  M  O  J  C  \  J  C  M  C  >  1  C  >  I  C  J  C  O  C  O  C  O  Boxplot for the resid. of log(H+) (83-86, monthly median, clust 3) CO  |  * N o > - N i n c o N ^ O ) 0 ^ i n i D N o i j i - N ( O O t - * i o N f f l o n c o N n c \ i n ^ m o o i x i T - n i n a j ( i ) O r - N n o ) o ) ' t ooT- -T-T-i-T-(\iwp)P)p)ncocomiD(0(DSNSNSS(OtO(ONNininininin«)NNNSNco(Dcoo)c it'0 ,  T  O O O O O O O O O O O O O O O O O O O O O O O O O O ^ T - ^ station  Figure 6.2.15(b)  Summary of the Effects and Residuals from Median Polish of log(H+) (83-86,monthly volume weighted mean.clust 1)  i  i  Yr  Sit  MonYr Figure 6.2.16(a)  MonSit  YrSit  Summary of the Effects and Residuals from Median Polish of log(H+) (83-86,monthly median,clust 1) i  i  i  i  Mon  Yr  Sit  MonYr Figure 6.2.16(b)  MonSit  YrSit  Summary of the Effects and Residuals from Median Polish of log(H+) (83-86,monthly volume weighted mean,clust 2)  Mon  Yr  Sit  MonYr Figure 6.2.17(a)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(H+) (83-86,monthly median,clust 2)  1  T  J ,  i  L  I  Mon  Yr  Sit  MonYr Figure  6.2.17(b)  MonSit  YrSit  Summary of the Effects and Residuals from Median Polish of log(H+) (83-86,monthly volume weighted mean.clust 3)  i i  Mon  Yr  Sit  MonYr  Figure 6.2.18(a)  MonSit  YrSit  res  Summary of the Effects and Residuals from Median Polish of log(H+) (83-86,monthly median,clust 3)  Sit  MonYr Figure  6.2.18(b)  MonSit  YrSit  res  Trend of log(H+) from 1983 to 1986 at the 86 Stations (calculated by monthly volume weighted mean)  0  no trend  -  down trend  +  up trend  Trend of log(H+) from 1983 to 1986 at the 86 Stations (calculated by monthly median)  Figure 6.2.19(b)  0  no trend  -  down trend  +  up trend  Trend of log(H+) from 1983 to 1986 in the USA (calculated by Kriging from monthly volume weighted mean)  -  down trend  +  up trend  Trend of log(H+) from 1983 to 1986 in the U S A (calculated by Kriging from monthly median)  -  down trend  +  up trend  TABLE  6.1.1  RESULTS OF MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF HYDROGEN ION ( M o n t h l y Volume W e i g h t e d Mean, '80 - '86) site ID  z  004a 010a 011a 017a 020a 021a 022a 023a 030a 031a 032a 034a 036a 038a 039a 040a 041a 049a 051a 052a 053a 055a 056a 058a 059a 064a 074a 075a 076a 168a 171a 173a  - 1 . 610 0 . 944 1 .275 0 . 961 - 1 .7 68 - 1 .231 - 1 .784 - 2 .186 - 0 .877 - 1 .115 -0 .169 - 0 .393 0 .769 - 0 .213 - 1 .432 - 0 . 654 - 0 .724 - 0 .731 -2 .477 - 2 .193 - 1 .106 - 1 .753 -0 .387 -0 .159 2 . 953 0 .066 1 .385 0 .337 - 1 .288 - 1 .042 -0 .457 -0 .469  p-value ] 0 .107 0 .345 0 .202 0 .336 0 .077 0 .218 0 .074 0 .029 0 .381 0 .265 0 .866 0 .694 0 .442 0 .832 0 .152 0 .513 0 .469 0 .465 0 .013 0 .028 0 .269 0 .080 0 .699 0 .874 0 .003 0 . 948 0 .166 0 .736 0 .198 0 .297 0 .648 0 .639  est'd slope  L-bd  - 0 .004 - 0 ..007 0 .002 - 0 ..001 0 .003 0. .0 0 .002 0 .0 . - 0 .002 - 0 ..004 - 0 .002 - 0 ..004 - 0 .002 - 0 . .004 - 0 .003 - 0 ..005 - 0 .002 - 0 . .005 - 0 .002 - 0 . .004 0 .0 - 0 . .001 - 0 .001 - 0 . .004 0 .001 - 0 . .001 0 .0 - 0 ..007 - 0 .002 - 0 . .005 - 0 .001 - 0 . .003 - 0 .001 . - 0 ..003 - 0 .001 - 0 . .004 - 0 .005 - 0 . .009 - 0 .004 - 0 . .007 - 0 .003 - 0 . .006 - 0 .002 - 0 . .003 0 .0 - 0 . .001 0 .0 - 0 . .001 0 .005 0. .002 0 .0 - 0 . .001 0 .002 0. .0 0 .0 - 0 . .001 - 0 .002 - 0 ..006 - 0 .002 - 0 . .004 0 .0 - 0 . .003 - 0 .001 - 0 . .004  245  80 %  U-b -0 .001 0 .007 0 .007 0 .005 - 0 .001 0 .0 0 .0 - 0 .001 0 .001 0 .0 0 .001 0 .002 0 .004 0 .005 0 .0 0 .001 0 .001 0 .001 - 0 .002 -0 .001 0 .0 0 .0 0 .001 0 .001 0 .007 0 .002 0 .004 0 .002 0 .0 0 .0 0 .001 0 .002  TABLE 6.1.2 RESULTS OF MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF HYDROGEN ION (Monthly Median, '80 - '8 6) site ID 004a 010a 011a 017a 020a 021a 022a 023a 030a 031a 032a 034a 036a 038a 039a 040a 041a 049a 051a 052a 053a 055a 056a 058a 059a 064a 074a 075a 076a 168a 171a 173a  z -1 .462 0 .191 1 . 656 1 . 639 -0 . 410 -1 .419 -1 .923 -2 .012 -0 .466 -1 .127 0 .733 0 .261 1 .605 -0 .240 -0 .259 -1 .021 -0 .894 -0 . 917 -1 .670 -1 .595 -0 .323 -0 .205 0 .282 -0 .569 4 . 155 -0 . 604 -0 .008 -0 .553 -0 .323 -1 .271 -0 .629 0 .373  p-value 0 .144 0 .849 0 .098 0 .101 0 . 682 0 .156 0 .054 0 .044 0 .641 0 .260 0 .464 0 .794 0 .109 0 .810 0 .795 0 .307 0 .372 0 .359 0 .095 0 .111 0 .747 0 .837 0 .778 0 .570 0 .0 0 .546 0 .993 0 .580 0 .747 0 .204 0 .529 0 .709  est'd slope -0,.004 0,.0 0..007 0 .003 , 0 .0 . -0,.002 -0,.003 -0..003 -0,.001 -0..002 0,.001 0,.0 0,.004 0 .0 , 0,.0 -0,.001 -0,.001 -0 .001 , -0,.004 -0..003 0 .0 , 0 .0 . 0..0 0,.0 0 .009 . 0 .0 . 0..0 0..0 0 .0 . -0..003 -0.,001 0..001  80 % ;  L-bd -0 .008 -0 .004 0 .001 0 .001 -0 .003 -0 .005 -0 .006 -0 .005 -0 .005 -0 .004 -0 .001 -0 .003 0 .001 -0 .008 -0 .003 -0 .004 -0 .004 -0 .004 -0 .007 -0 .006 -0 .004 -0 .001 -0 .001 -0 .002 0 .006 -0 .002 -0 .001 -0 .002 -0 .005 -0 .005 -0 .003 -0 .002  246  U-b  0 .0 0 .007 0 .012 0 .006 0 .001 0 .0 -0 .001 -0 .001 0 .001 0 .0 0 .003 0 .004 0 .006 0 .005 0 .002 0 .0 0 .001 0 .0 0 .0 0 .0 0 .003 0 .001 0 .002 0 .001 0 .012 0 .001 0 .002 0 .001 0 .002 0 .0 0 .001 0 .005  TABLE 6.2.1 RESULTS OF MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF HYDROGEN ION (Monthly Volume Weighted Mean, '83 - '86) site ID 004a 007a 010a 011a 012a 015a 016a 017a 020a 021a 022a 023a 024a 025a 028a 029a 030a 031a 032a 033a 034a 035a 036a 037a 038a 039a 040a 041a 046a 047a 049a 051a 052a 053a 055a 056a 058a 059a 061a 062a 063a 064a 0 65b 068a 070a 071a 073a  Z -1..926 -0..324 1..769 1..609 0,. 947 0..670 -1..246 3,.918 0 .101 , 0 .204 , 0 .436 , 0 .169 , -1 .502 -0..240 1,.526 -0 .542 , 1,.559 -0..367 -1,.577 -1,.822 -2 .708 -1 .507 2 .729 0 .956 -2 .568 0 .898 . 1,.271 0 .391 , 1..395 1 . 841 0..880 1,.155 -0,.578 2 .446 . 1..307 2 .195 . 0 .613 , 1,.155 2..301 -0..335 2.. 146 1..253 -0..147 -0 . ,294 -0 . .142 -0 . .773 1.. 683  p-value 0 .054 0 .746 0 . 077 0 . 108 0 .344 0 .503 0 .213 0 .0 0 . 920 0 .838 0 . 663 0 .866 0 .133 0 .810 0 .127 0 .588 0 .119 0 .714 0 .115 0 .068 0 .007 0 .132 0 .006 0 .339 0 .010 0 .369 0 .204 0 .696 0 .163 0 .066 0 .379 0 .248 0 .564 0 .014 0 .191 0 .028 0 .540 0 .248 0 .021 0 .737 0 .032 0 .210 0 .883 0 .769 0 .887 0 .439 0 .092  est'd slope -0 .011 -0 .002 0 .007 0 .009 0 .008 0 .001 -0 .009 0 .016 0 .0 0 .0 0 .001 0 .001 -0 .008 -0 .001 0 .006 -0 .002 0 .006 -0 .002 -0 .004 -0 .005 -0 .016 -0 .022 0 .012 0 .002 -0 .028 0 .005 0 .005 0 .001 0 .003 0 .006 0 .004 0 .007 -0 .003 0 .009 0 .004 0 .006 0 .002 0 .003 0 .005 -0 .003 0 .005 0 .003 0 .0 -0 .002 -0 .001 -0 .007 0 .007  80 % !  L-bd -0 .019 -0 .011 0 .002 0 .003 -0 .004 -0 .001 -0 .019 0 .009 -0 .003 -0 .004 -0 .003 -0 .003 -0 .014 -0 .005 0 .001 -0 .008 0 .002 -0 .006 • -0 .007 -0 .009 -0 .023 -0 .036 0 .006 -0 .001 -0 .042 -0 .002 0 .0 -0 .003 0 .0 0 .002 -0 .002 0 .0 -0 .009 0 .003 0 .0 0 .003 -0 .002 0 .0 0 .002 -0 .016 0 .002 0 .0 -0 .003 -0 .014 -0 .008 -0 . 020 0 .002  247  U-bd  -0 .004 0 .004 0 .014 0 .016 0 .017 0 .007 0 .0 0 .021 0 .003 0 .003 0 .007 0 .005 -0 .002 0 .002 0 .012 0 .004 0 .012 0 .002 -0 .001 -0 .002 -0 .009 -0 .002 0 .019 0 .006 -0 .014 0 .012 0 .009 0 .005 0 .005 0 .009 0 .011 0 .014 0 .003 0 .016 0 .007 0 .010 0 .005 0 .007 0 .008 0 .008 0 .008 0 .006 0 .003 0 .008 0 .008 0 .005 0 .013  TABLE  6 . 2 . 1 (continued)  RESULTS OF MANN-KENDALL T E S T S AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF HYDROGEN ION ( M o n t h l y V o l u m e W e i g h t e d Mean, ' 8 3 - ' 8 6 ) site ID 074a 075a 076a 077a 078a 160a 161a 163a 164a 166a 168a 171a 172a 173a 249a 251a 252a 253a 254a 255a 257a 258a 268a 271a 272a 273a 275a 277a 278a 279a 280a 281a 282a 283a 285a 339a 349a 350a 354a  Z  3 .136 3 . 920 1 .027 -1 .155 -0 .685 1 .096 -0 .382 0 .871 0 .161 0 . 624 0 . 994 2 .755 1 .311 0 .578 1 .448 1 . 976 -1 .075 0 .862 -1 .240 0 .453 1 .922 0 .265 0 .791 -0 .792 -1 .733 -2 .106 0 . 933 1 . 687 0 .744 2 .160 1 . 129 2 .714 1 .698 -1 .467 1 .316 1 .350 2 .036 2 .759 0 .450  F>-value 0. 002 0. 0 0 .304 0. 248 0. 493 0 .273 0 .702 0 .384 0. 872 0. 533 0 .320 0 .006 0. 190 0. 563 0. 148 0. 048 0. 282 0 .389 0. 215 0 .650 0 .055 0. 791 0 .429 0 .428 0 .083 0. 035 0. 351 0. 092 0 .457 0. 031 0 .259 0. 007 0 .090 0. 143 0. 188 0. 177 0. 042 0. 006 0. 653  est' d slope 0,.008 0 .009 , 0 .005 . -0 .010 -0,.004 0..007 -0 .001 0..003 0 .001 0 .004 , 0..005 0..009 0 .003 , 0,.003 0..008 0 .013 . -0 .007 0 .003 -0 .005 0 .004 0 .009 0..001 0 .005 . -0 , .003 -0,.004 -0..011 0,.004 0,.006 0..009 0,.011 0,.006 0,.010 0 .007 , -0,.010 0..007 0..014 0..011 0..012 0..002  80% L-bd 0 .005 0 .006 -0 .002 -0 .023 -0 .014 -0 .001 -0 .007 -0 .002 -0 .007 -0 .003 -0 .001 0 .005 0 .0 -0 .004 0 .0 0 .005 -0 .013 -0 .002 -0 .012 -0 .010 0 .003 -0 .004 -0 .003 -0 .008 -0 .008 -0 .017 -0 .002 0 .001 -0 .003 0 .005 -0 .001 0 .005 0 .002 -0 .021 0 .0 0 .0 0 .005 0 .006 -0 .004  248  U-bd 0. 012 0 .012 0 .011 0. 001 0 .005 0 .013 0 .003 0 .008 0. 008 0 .010 0 .010 0 .014 0. 008 0 .008 0. 015 0. 023 0. 002 0 .009 0. 0 0. 014 0. 016 0 .007 0. 012 0 .002 -0. 001 -0. 005 0. 010 0. 010 0. 029 0. 017 0. 013 0. 015 0. 014 - 0 . 001 0. 015 0. 027 0. 016 0. 017 0. 008  RESULTS OF MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF HYDROGEN ION (Monthly M e d i a n , '83 - '86) site ID 004a 007a 010a 011a 012a 015a 016a 017a 020a 021a 022a 023a 024a 025a 028a 029a 030a 031a 032a 033a 034a 035a 036a 037a 038a 039a 040a 041a 046a 047a 049a 051a 052a 053a 055a 056a 058a 059a 061a 062a 063a 064a 065b 068a 070a 071a 073a  Z  p-value  -1 . 825 0 .0 0 .382 1 .324 0 .379 0 .859 -0 .743 3 .163 1 .770 -0 .356 -0 .791 0 .009 -0 .178 -1 .093 1 .174 -0 .595 1 .247 -0 . 908 -1 .706 -0 . 613 -1 .875 -1 .790 1 . 946 1 . 657 -2 .265 2 .373 1 .040 1 .067 1 .102 1 .740 1 .004 1 .541 1 . 402 2 .563 1 . 698 1 . 929 0 .631 1 . 954 0 . 985 0 .084 0 . 688 0 .755 0 .084 0 . 142 0 .718 -0 . 418 2 .622 1  0 .068 1 .000 0 .702 0 .185 0 .705 0 .390 0 .457 0 .002 0 .077 0 .722 0 .429 0 .993 0 .859 0 .274 0 .240 0 .552 0 .212 0 .364 0 .088 0 .540 0 .061 0 .074 0 .052 0 .097 0 .023 0 .018 0 .298 0 .286 0 .270 0 .082 0 .315 0 .123 0 .161 0 .010 0 .090 0 .054 0 .528 0 .051 0 .325 0 .933 0 .492 0 .450 0 .933 0 .887 0 .472 0 .676 0 .009  80 %  est'd slope -0 .014 0 .0 0 .003 0 .010 0 .005 0 .005 -0 .004 0 .012 0 .004 -0 .002 -0 .003 0 .0 -0 .001 -0 .005 0 .008 -0 .003 0 .007 -0 .004 -0 .006 -0 .001 -0 .016 -0 .022 0 .009 0 .006 -0 .029 0 .013 0 .004 0 .003 0 .002 0 .004 0 .005 0 .008 0 .007 0 .015 0 .005 0 .006 0 .002 0 .008 0 .002 0 .0 0 .002 0 .002 0 .0 0 .002 0 .007 -0 .007 0 .010  :  L-bd -0 .022 -0 .011 -0 .008 0 .0 -0 .010 -0 .002 -0 .012 0 .007 0 .001 -0 .011 -0 .008 -0 .005 -0 .007 -0 .009 0 .0 -0 .010 -0 .001 -0 .009 -0 .012 -0 .005 -0 .026 -0 .035 0 .003 0 .001 -0 .042 0 .007 -0 .001 -0 .001 -0 .001 0 .001 -0 .001 0 .001 0 .0 0 .007 0 .001 0 .001 -0 .002 0 .003 -0 .001 -0 .013 -0 .001 -0 .001 -0 .002 -0 .011 -0 .006 -0 .019 0 .004  249  U-b< -0 .004 0 .008 0 .013 0 .019 0 .018 0 .011 0 .003 0 .018 0 .007 0 .005 0 .002 0 .004 0 .005 0 .0 0 .017 0 .003 0 .012 0 .001 -0 .002 0 .002 -0 .005 -0 .009 0 .015 0 .011 -0 .013 0 .019 0 .008 0 .008 0 .005 0 .008 0 .011 0 .014 0 .012 0 .022 0 .008 0 .010 0 .005 0 .013 0 .006 0 .014 0 .005 0 .005 0 .002 0 .012 0 .020 0 .006 0 .016  TABLE  6.2.2  (continued)  RESULTS OF MANN-KENDALL TESTS AND SLOPE ESTIMATES FOR THE CONCENTRATIONS OF HYDROGEN ION (Monthly M e d i a n , '83 - '86) site ID 074a 075a 076a 077a 078a 160a 161a 163a 164a 166a 168a 171a 172a 173a 249a 251a 252a 253a 254a 255a 257a 258a 268a 271a 272a 273a 275a 277a 278a 279a 280a 281a 282a 283a 285a 339a 349a 350a 354a  Z  2 .210 3 .013 0 .0 -0 .807 -0 .558 -0 .724 -0 .613 0 .843 1 .790 0 .990 1 .846 2 .480 1 .820 1 .395 2 .486 2 . 955 -0 .800 1 .752 -0 .123 1 .182 1 . 477 1 .004 1 .155 -0 .039 -1 .555 -0 .729 0 .907 1 .834 1 .772 1 .431 1 .715 2 .421 1 .058 -0 . 622 1 .013 2 .397 1 .790 2 .407 1 .780  Fi - v a l u e 0 .027 0. 003 1. 000 0 .420 0 .577 0 .469 0 .540 0 .399 0 .074 0. 322 0. 065 0. 013 0 .069 0 .163 0 .013 0. 003 0 .424 0 .080 0 .902 0 .237 0 .140 0 .316 0. 248 0. 969 0 .120 0. 466 0. 365 0 .067 0 .076 0. 152 0 .086 0 .015 0. 290 0 .534 0. 311 0. 017 0. 074 0 .016 0 .075  80 %  est'd slope 0 .008 0 .008 0 .0 -0 .010 -0 .004 -0 .004 -0 .003 0 .004 0 .011 0 .005 0 .008. 0 .010 0 .004 0 .009 0 .011 0 .023 -0 .003 0 .007 -0 .001 0 .009 0 .008 0 .006 0 .007 0 .0 -0 .006 -0 .007 0 .006 0 .007 0 .015 0 .008 0 .013 0 .012 0 .003 -0 .007 0 .005 0 .020 0 .010 0 .011 0 .014  1  L-bd 0 .003 0 .005 -0 .011 -0 .021 -0 .016 -0 .013 -0 .009 -0 .003 0 .003 -0 .002 0 .003 0 .005 0 .001 0 .001 0 .004 0 .014 -0 .011 0 .002 -0 .010 -0 .001 0 .001 -0 .001 -0 .001 -0 .005 -0 .012 -0 .015 -0 .003 0 .002 0 .004 0 .001 0 .002 0 .006 0 .0 -0 .022 -0 .001 0 .009 0 .002 0 .005 0 .001  250  U-b 0 .013 0 .012 0 .012 0 .005 0 .003 0 .004 0 .003 0 .011 0 .020 0 .015 0 .015 0 .014 0 .007 0 .016 0 . 021 0 .030 0 .002 0 .013 0 .011 0 .019 0 .014 0 .014 0 .015 0 .005 -0 .001 0 .003 0 .014 0 .012 0 .032 0 .019 0 .024 0 .019 0 .008 0 .008 0 .012 0 .031 0 .016 0 .016 0 .026  Chapter 7  S U M M A R Y A N D F U R T H E R STUDIES  The primary purpose of this study is to detect and estimate the possible temporal trends in different levels of chemical constituents of acid deposition at different locations. Spatial patterns and seasonal patterns of the levels of the chemical concentrations are also of interests. Three ions, sulphate, nitrate and hydrogen ion are analyzed in this study. Some characteristics of the chemical precipitation data, for example, nonnormality, existence of missing data and the limited number of observations available, create some difficulties in using the traditional parametric statistical methods.  Some nonparametric  statistical techniques are thus suggested and used here, namely, hierarchical clustering, median polishing, the Mann-Kendall Test for monotone trend, Sen's slope estimate and Kriging. The data are clustered into two to three clusters according their temporal patterns, then transformed by the log transformation so that the transformed data are approximately symmetric.  The analyses are then based on the transformed data. The result of median  polish suggests that there is a V-shaped trend for all the three chemicals in the subregion of highest industrial concentration. That is, the concentrations of the chemicals decreased first and then increased (or remained constant in a few cases) during the period, 1980 1986, with the change point around 1983.  Strong seasonality and spatial patterns are  discovered as well where the seasonal patterns of sulphate and hydrogen ion are similar but different from that of nitrate.  The Mann-Kendall Test for monotone trend and Sen's slope estimation procedure are applied to the deseasonalized data for each site. The main result is that there is overall an up-trend in the East for all the three ions during the period of 1983 - 1986. 251  This result is  consistent  with the results of median polish.  Sen's nonparametric slope estimate is  obtained for each site. Based on these estimates, the slope estimate is obtained by Kriging interpolation for each integer degree grid point of longitude and latitude across the 4 8 conterminous states in the United States. For further study, the outliers found in the data need to be examined; the serial independence of the data may be examined by testing independence to assure the validity of the trend test; and a rank test for umbrella alternatives can be used to test the V-shaped trend for which the monotone trend test is not valid. In addition, that the spatial patterns of the estimated trends for sulphate and nitrate are different from that for hydrogen ion suggests the possibility of multivariate analysis to explore the relationship of the concentration levels of different chemicals.  252  BIBLIOGRAPHY Altwicker, E . R. and Johannes, A . H . (1987).  Spatial and Historical Trends in acidic  deposition: A Graphical Intersite Comparison, Atmospheric Environment,Vol.  21 No. 1,  pp.129-137. Bilonick, R. A . (1985).  The Space-Time Distribution of Sulphate Deposition in the  Northeastern United States, Atmospheric Environment,Vol. 19 No. 11, pp.1829-1845. Bilonick, R. A. (1987). Monthly Hydrogen Ion Deposition Maps for the Northeastern U.S. from July, 1982 to September, 1984,  Consolidation Coal Company, Coal Plaza, Pittsburgh,  PA 15241. Bradley, J. V . (1968). Distribution-Free Statistical Tests, Prentice-Hall, Englewood Cliffs, NJ. Dana, M . T. and Easter, R . C . (1987). Precipitation  Chemistry from the  Statistical Summary and Analyses of Event  MAP3S  Network,  1976-1983,  Atmospheric  Environment,Vol. 21 No. 1, pp.113-128. Delfiner, P. (1965). Linear Estimation of Non Stationary Spatial Phenomena,  Advanced  Geostatitics in the Mining Industry, Reidel, Boston. Devary, J. L . and Doctor, P.D. (1981). Geostatistical Modeling of Pore Velocity, BattellePacific Northwest Laboratory, Report #PNL-3789. Devary, J. L . and Rice W . A . (1982). Geosciences Research  Geostatistics Software User's Manual for the  and Engineering  11/70  Computer System,  Battelle-Pacific  Northwest Laboratory Report. Dietz, E . J. and Killeen, T. J. (1981).  A Nonparametric Multivariate Test for Monotone  Trend With Pharmaceutical Applications, Journal of the American Statistical Association, Vol. 76, No. 373, pp 169-174 Egbert, G . D. and Lettenmaier, D. P. (1986).  Stochastic Modeling of the Space-Time  Structure of Atmospheric Chemical Deposition,  Water Resources Research, Vol. 22, No. 2,  pp. 165-179  253  Eynon, B. P. and Switzer , P. (1983). The variability of rainfall acidity, The Canada Journal of Statistics, Vol. 11, No. 1, pp. 25-49. Farrell, R. (1980). Methods for Classifying Changes in Environmental Conditions, Tech. report VRF-EPA7. 4-FR80-1. Vector Research Inc., Ann Arbor, Mich. Finkelstein, P. L . (1984). The spatial analysis of acid precipitation data, Journal of Climate and Applied Meteorology, Vol. 23, pp. 52-62. Gilbert, R. O. (1987).  Statistical Methods for Environmental Pollution Monitoring, New  York: Van Nostrand Reinhold. Hirsch R. M . and Slack, J. R. (1984). A Nonparamatric Trend Test for Seasonal Data With Serial Dependence, Water Resources Research, Vol. 20, No. 6, pp727-732. Hirsch R. M . , Slack, J. R., and Smith R. A . (1982). Techniques of Trend Analysis for Monthly Water Quality Data, Water Resources Research, Vol. 18, No. 1, pp. 107-121. Interagancy Task Force on Acid Precipitation. (1982).  National Acid Precipitation  Assessment Plan. Jernigan, R. W. (1986). A Primer on Kriging, report for U . S. Environmental Protection Agency, Washington, D . C . Johnson, R. A . , Wichern, D.W. (1982).  Applied  Multivariate  Statistical  Analysis.  Prentice-Hall, Englewood Cliffs, N J . Jonckheere, A . R. (1954).  A distribution-free k sample test against ordered alternatives,  Biometrika, Vol. 41, pp. 135-145. Kendall, M . G . (1970). Rank Correlation Methods, London: Charles Griffin. Kendall, M . G. (1975). Rank Correlation Methods, 4th ed. London: Charles Griffin. Le, D. N. and Petkau, A . J. (1988), The variability of rainfall acidity revisited, The Canadian Journal of Statistics, Vol. 16, No. 1, pp.15. Mann, H . B. (1945) Non-parametric tests against trend, Econometrica, Vol. 13,pp. 245-259.  254  Mardia, K. V . , Kent, J. T., Bibby, J. M . (1979). Multivariate Analysis. London: Academic Press. Matheron, G . (1973). The Intrinsic Random Functions and Their Applications, Advances in Applied Probability, Vol. 5, pp. 439-468. NADP.  (1980). N A D P Quality assurance Report, Central Analytical Laboratory, 1/1/79 to  12/31/79. Colorado State University, Ft., Collins, C O . NADP.  (1982).  National Acid Deposition Program Instruction Manual, Site Operation,  natural Resource Ecology Laboratory, Colorado State University, Ft., Collins, C O . NADP.  (1984a). Instruction Manual, N A D P / N T N Site Selection and Installation,  natural  Resource Ecology Laboratory, Colorado State University, Ft., Collins, C O . N A D P . (1984b). N A D P Quality assurance Plan, Deposition Monitoring, natural Resource Ecology Laboratory, Colorado State University, Ft., Collins, C O . Neuman, S. H . and Jacobson, E.A. (1984). Analysis of Nonintrinsic Spatial Variability by Residual Kriging with Application to Regional Groundwater Levels, Mathematical Geology, Vol. 16, No, 5, pp. 499-521. Olsen, A . R . and Slavich A . L . (1986). Acid Precipitation in North America: 1984 Annual Data Summary from Acid Deposition System Data Base.  U.S. Environmental Protection  Agency, Research Triangle Park, NC. EPA/600/4-86/033. Robertson, J. K . and Wilson, J. W. (1985).  Design of the National Trends Netwok for  Monitoring the Chemistry of Atmospheric Precipitation. U.S. Geological Survey Circular 964 U . S. Geological Survey ; Alexandria, V A . Schertz, T. L . and Hirsch, R. M . (1985).  Trend Analysis of weekly Acid Rain Data —  1978-83, U.S. Geological Survey, Water Resources Investigations Report 85--4211 Seilkop, S. K . and Finkelstein, P. L . (1987).  Acid Precipitation Patterns and Trends in  Eastern North America, 1980-84, Journal of Climate and Applied Meteorology, Vol. 26, pp. 980-994. Sen, P. K.(1968a).  On a class of aligned rand order tests in two-way layouts, Journal of  the American Statistical Association, Vol. 39, pp. 1115-1124  255  Sen, P. K.(1968b). Estimates of the regression coefficient based on Kendall's tau, Journal of the American Statistical Association, Vol. 63, pp. 1379-1389. Tukey, J. W. (1977). Exploratory Data Analysis, Addison-Wesley, Reading, Mass. van Belle, G . and Hughes, J. P. (1984). Nonparametric Tests for Trend in Water Quality, Water Resources Research, Vol. 20, No. l,pp. 127-136.  256  

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