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Preliminary Spatial Data Analysis

In this example, the 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.

It is instructive to see the locations of the measured points in the area where you want to perform spatial simulations. It is generally desirable to have these locations scattered evenly around the simulation area.

First, the data are input and the sample locations plotted.

   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
      ;

   proc gplot data=thick;
      title 'Locations of Measured Samples';
      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;

   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 70.1 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. In fact, a reasonable approximation of the spatial process generating the coal seam data is given by

where the is a Gaussian SRF with Gaussian covariance structure

C_z(h) = c₀exp(-[(h²)/(a₀²)])

Note that the term "Gaussian" is used in two ways in this description. For a set of locations s₁,s₂, ... ,s_n, the random vector

has a multivariate Gaussian or normal distribution .The (i,j)th element of is computed by C_z(s_i-s_j), which happens to be a Gaussian functional form. Any functional form for C_z(h) yielding a valid covariance matrix can be used. Both the functional form of C_z(h) and the parameter values