Preliminary Spatial Data Analysis

The VARIOGRAM Procedure

Preliminary Spatial Data Analysis

The simulated data consist of coal seam thickness measurements (in feet) taken over an approximately square area. The coordinates are offsets from a point in the southwest corner of the measurement area, with the north and east distances in units of thousands of feet.

First, the data are input.

   data thick;
      input east north thick @@;
      datalines;
       0.7  59.6  34.1   2.1  82.7  42.2   4.7  75.1  39.5 
       4.8  52.8  34.3   5.9  67.1  37.0   6.0  35.7  35.9
       6.4  33.7  36.4   7.0  46.7  34.6   8.2  40.1  35.4   
      13.3   0.6  44.7  13.3  68.2  37.8  13.4  31.3  37.8
      17.8   6.9  43.9  20.1  66.3  37.7  22.7  87.6  42.8 
      23.0  93.9  43.6  24.3  73.0  39.3  24.8  15.1  42.3
      24.8  26.3  39.7  26.4  58.0  36.9  26.9  65.0  37.8 
      27.7  83.3  41.8  27.9  90.8  43.3  29.1  47.9  36.7
      29.5  89.4  43.0  30.1   6.1  43.6  30.8  12.1  42.8
      32.7  40.2  37.5  34.8   8.1  43.3  35.3  32.0  38.8
      37.0  70.3  39.2  38.2  77.9  40.7  38.9  23.3  40.5
      39.4  82.5  41.4  43.0   4.7  43.3  43.7   7.6  43.1
      46.4  84.1  41.5  46.7  10.6  42.6  49.9  22.1  40.7
      51.0  88.8  42.0  52.8  68.9  39.3  52.9  32.7  39.2
      55.5  92.9  42.2  56.0   1.6  42.7  60.6  75.2  40.1
      62.1  26.6  40.1  63.0  12.7  41.8  69.0  75.6  40.1
      70.5  83.7  40.9  70.9  11.0  41.7  71.5  29.5  39.8
      78.1  45.5  38.7  78.2   9.1  41.7  78.4  20.0  40.8
      80.5  55.9  38.7  81.1  51.0  38.6  83.8   7.9  41.6
      84.5  11.0  41.5  85.2  67.3  39.4  85.5  73.0  39.8 
      86.7  70.4  39.6  87.2  55.7  38.8  88.1   0.0  41.6
      88.4  12.1  41.3  88.4  99.6  41.2  88.8  82.9  40.5 
      88.9   6.2  41.5  90.6   7.0  41.5  90.7  49.6  38.9 
      91.5  55.4  39.0  92.9  46.8  39.1  93.4  70.9  39.7 
      94.8  71.5  39.7  96.2  84.3  40.3  98.2  58.2  39.5
      ;

It is instructive to see the locations of the measured points in the area where you want to perform spatial prediction. It is desirable to have these locations scattered evenly around the prediction area. If this is not the case, the prediction error might be unacceptably large where measurements are sparse. The following GPLOT procedure is useful in determining potential problems:

   proc gplot data=thick;
      title 'Scatter Plot of Measurement Locations';
      plot north*east / frame cframe=ligr haxis=axis1 
                        vaxis=axis2; 
      symbol1 v=dot color=blue;
      axis1 minor=none;
      axis2 minor=none label=(angle=90 rotate=0);
      label east   = 'East'
            north  = 'North'
      ;
   run;

Figure 70.1: Scatter Plot of Measurement Locations

As Figure 58.2 indicates, while the locations are not ideally spread around the prediction area, there are not any large areas lacking measurements. You now can look at a surface plot of the measured variable, the thickness of coal seam, using the G3D procedure. This is a crucial step. Any obvious surface trend has to be removed before you compute and estimate the model of spatial dependence (the semivariogram model).

   proc g3d data=thick;
      title 'Surface Plot of Coal Seam Thickness';  
      scatter east*north=thick / xticknum=5 yticknum=5
         grid zmin=20 zmax=65;
      label east  = 'East'
            north = 'North'
            thick = 'Thickness'
      ;
   run;

Figure 70.2: Surface Plot of Coal Seam Thickness

Figure 58.1 shows the small-scale variation typical of spatial data, but there does not appear to be any surface trend. Hence, you can work with the original thickness data rather than residuals from a trend surface fit.

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