The variability of , as modeled by

with the Gaussian covariance structure found previously, is not obvious from the covariance model form and parameters. The variation around the mean of the surface is relatively small, making it difficult visually to pick up differences in surface plots of simulated realizations.

Instead, you can compute the mean for each location on a grid from a series of realizations in a simulation. Then, the standard deviation of all the simulated values at each grid location will provide you with a measure of the variability of for the given covariance structure. You can also investigate variations at selected grid points in more detail, as shown in the Variability at Selected Locations.

In the present example you will see how to use ODS Graphics with PROC SIM2D to investigate the mean and standard deviation of simulated values. In particular, you produce 5,000 realizations of a simulation with PROC SIM2D, where you specify the Gaussian model with the parameters found previously. You want the simulated data to pass through the simulated values, so first you define the data with the following DATA step:

   data thick;
      input East North Thick @@;
      label Thick='Coal Seam Thickness';
      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
      55.8  50.5  38.1  96.2  84.3  40.3  98.2  58.2  39.5
      ;

Since this is a conditional simulation, you can specify the OBSERV option in the PLOT option in PROC SIM2D to see the locations and values of the measured points in the area where you want to perform spatial simulations.

Furthermore, the MEANS suboption in the PLOT option specifies that you want to create a plot that shows the means of the simulated values across the region. The MEANS suboption with no other arguments produces a plot that shows the contours of the simulated means in the foreground and the gradient of the simulated standard deviations in the background.

You obtain these PROC SIM2D results at the nodes of an output grid that you specify according to your application needs. In the present analysis, a convenient area that encompasses all the Thick data points is a square with a side length of 100,000 feet. You define a regular grid for your simulation in this area. Assume a distance of 2,500 feet between grid nodes in both directions for a smooth contour plot. Based on this choice, your square grid has 41 nodes on each side. This means that PROC SIM2D computes the simulated values at a total of 1,681 grid points. You use the GRID statement of the PROC SIM2D to specify this grid.

The SIMULATE statement specifies the parameters of your simulation across the output grid. In particular, the VAR= option specifies the conditional simulation variable. The number of realizations in the simulation is specified with the NUMREAL option. The SEED= option specifies the seed for the simulation random number generator.

The spatial correlation model for the simulation is also specified in the SIMULATE statement. You specify the model type by using the FORM= option. The options SCALE= and RANGE= specify the covariance structure sill and range parameters, respectively, as discussed in the previous section.

Although it is not included in the original spatial structure, note that a minimal nugget effect is specified with the NUGGET= option to avoid singularity issues. Singularity can appear in the present example as a result of the combined use of the Gaussian covariance model and relatively short distances between nodes, data, or nodes and data in the simulation area.

These steps are implemented using the following DATA step and statements:

   ods graphics on;

   proc sim2d data=thick outsim=sim plot=(observ means);
      coordinates xc=East yc=North;
      simulate var=Thick numreal=5000 seed=79931
         scale=7.2881 range=30.6239 nugget=1e-8 form=gauss;
      mean 40.1173;
      grid x=0 to 100 by 2.5 y=0 to 100 by 2.5;
   run;

The table in Figure 79.1 shows the number of observations read and used in the conditional simulation. This table can provide you with useful information in case you have missing values in the input data.