Getting Started

Spatial simulation, just like spatial prediction, requires a model of spatial dependence, usually in terms of the covariance C _z ( h ). For a given set of spatial data Z ( s _i ) ,i = 1 , ,n , the covariance structure (both the form and parameter values) can be found by the VARIOGRAM procedure. This example uses the coal seam thickness data that is also used in the 'Getting Started' section of Chapter 80, 'The VARIOGRAM Procedure.'

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;  

Figure 65.1: Locations of Measured Samples

  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 65.2 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

Figure 65.2: Surface Plot of Coal Seam Thickness

where the µ ( s ) is a Gaussian SRF with Gaussian covariance structure

Note that the term 'Gaussian' is used in two ways in this description. For a set of locations s ₁ , s ₂ , , s _n , the random vector

has a multivariate Gaussian or normal distribution N _n ( µ, & pound ; ). The (i, j)th element of is computed by C _z ( s _i ˆ’ 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

• µ = 40 . 14

• c = 7 . 5

• a = 30 .

are visually estimated using PROC VARIOGRAM, a DATA step, and the GPLOT procedure. Refer to the 'Getting Started' section beginning on page 4852 in the chapter on the VARIOGRAM procedure for details on how these parameter values are obtained.

The choice of a Gaussian functional form for C _z ( h ) is simply based on the data, and it is not at all crucial to the simulation. However, it is crucial to the simulation method used in PROC SIM2D that Z ( s ) be a Gaussian SRF. For details, see the section 'Computational and Theoretical Details of Spatial Simulation' beginning on page 4106.

Investigating Variability by Simulation

The variability of Z ( s ),modeledby

with the Gaussian covariance structure C _z ( h ) 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 investigate variations at selected grid points.

To do this investigation, this example uses PROC SIM2D and specifies the Gaussian model with the parameters found previously. Five thousand simulations (iterations) are performed on two points: the extreme south-west point of the region and a point towards the north-east corner of the region. Because of the irregular nature of these points, a GDATA= data set is produced with the coordinates of the selected points.

Summary statistics are computed for each of these grid points by using a BY statement in PROC UNIVARIATE.

  data grid;   input xc yc;   datalines;   0   0   75  75   run;   proc sim2d data=thick outsim=sim1;   simulate var=thick numreal=5000 seed=79931   scale=7.5 range=30.0 form=gauss;   mean 40.14;   coordinates xc=east yc=north;   grid gdata=grid xc=xc yc=yc;   run;   proc sort data=sim1;   by gxc gyc;   run;   proc univariate data=sim1;   var svalue;   by gxc gyc;   title 'Simulation Statistics at Selected Grid Points';   run;  

  Simulation Statistics at Selected Grid Points   ------ X-coordinate of the grid point=0 Y-coordinate of the grid point=0 -------   The UNIVARIATE Procedure   Variable: SVALUE (Simulated Value at Grid Point)   Moments   N                        5000    Sum Weights               5000   Mean               40.1387121    Sum Observations    200693.561   Std Deviation      0.54603592    Variance            0.29815523   Skewness   0.0217334    Kurtosis   0.0519914   Uncorrected SS     8057071.54    Corrected SS          1490.478   Coeff Variation    1.36037231    Std Error Mean      0.00772211   Basic Statistical Measures   Location                    Variability   Mean     40.13871     Std Deviation            0.54604   Median   40.14620     Variance                 0.29816   Mode       .          Range                    3.81973   Interquartile Range      0.76236  

Figure 65.3: Simulation Statistics at Grid Point (XC=0, YC=0)

  Simulation Statistics at Selected Grid Points   ------ X-coordinate of the grid point=0 Y-coordinate of the grid point=0 -------   The UNIVARIATE Procedure   Variable: SVALUE (Simulated Value at Grid Point)   Tests for Location: Mu0=0   Test           -Statistic-    -----p Value------   Student's t    t  5197.892    Pr > t         <.0001   Sign           M      2500    Pr >= M   <.0001   Signed Rank    S   6251250    Pr >= S   <.0001   Quantiles (Definition 5)   Quantile      Estimate   100% Max       41.9369   99%            41.4002   95%            41.0273   90%            40.8334   75% Q3         40.5168   50% Median     40.1462   25% Q1         39.7544   10%            39.4509   5%             39.2384   1%             38.8656   0% Min         38.1172   Extreme Observations   ------Lowest-----        -----Highest-----   Value      Obs           Value      Obs   38.1172     2691         41.8085     1149   38.2959     1817         41.8251     3612   38.3370     3026         41.8446     3757   38.3834     2275         41.9338      135   38.4198     3100         41.9369     4536  

Figure 65.4: Simulation Statistics at Grid Point (XC=75, YC=75)