The SIM2D Procedure

Example 91.2 Variability at Selected Locations

This example exhibits a more detailed investigation of the variation of simulated `Thick` variable values. You use the same `thick` data set from the section Getting Started: SIM2D Procedure, and you are interested in the simulated values statistics at two selected grid points.

Specifically, you perform a simulation asking for 5,000 realizations (iterations) at two points of the region defined in the section Preliminary Spatial Data Analysis. These are the extreme southwest point and a point toward the northeast corner of the region. Since you want to avoid performing the simulation across the whole region, you need to produce a GDATA= data set to specify the coordinates of the selected points. These steps are implemented using the following DATA step and statements:

```title 'Investigation of Random Field Variability';

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
;
```
```data grid;
input xc yc;
datalines;
0   0
75  75
;
```

Then, you run PROC SIM2D with the same parameters and characteristics as those shown in the section Preliminary Spatial Data Analysis. This time, however, you ask for simulated values only at the two locations you specified in the previous DATA step. The following statements execute the requested simulation:

```proc sim2d data=thick outsim=sim1;
coordinates xc=East yc=North;
simulate var=Thick numreal=5000 seed=79931
scale=7.4599 range=30.1111 form=gauss;
mean 40.1173;
grid gdata=grid xc=xc yc=yc;
run;
```

After the simulation is performed, summary statistics are computed for each of the specified grid points. In particular, you use PROC UNIVARIATE and a BY statement to request the quantiles and the extreme observations at these locations, as the following statements show:

```proc sort data=sim1;
by gxc gyc;
run;

proc univariate data=sim1;
var svalue;
by gxc gyc;
ods select Quantiles ExtremeObs;
title 'Simulation Statistics at Selected Grid Points';
run;
```

The summary statistics for the first grid point (`East`=0, `North`=0) are presented in Output 91.2.1.

Output 91.2.1: Simulation Statistics at Grid Point (`East`=0, `North`=0)

 Simulation Statistics at Selected Grid Points

The UNIVARIATE Procedure
Variable: SVALUE (Simulated Value at Grid Point)

Quantiles (Definition 5)
Level Quantile
100% Max 42.4207
99% 41.8960
95% 41.5315
90% 41.3419
75% Q3 41.0324
50% Median 40.6701
25% Q1 40.2871
10% 39.9904
5% 39.7825
1% 39.4181
0% Min 38.6864

Extreme Observations
Lowest Highest
Value Obs Value Obs
38.6864 2691 42.2952 1149
38.8611 1817 42.3114 3612
38.9013 3026 42.3305 3757
38.9467 2275 42.4177 135
38.9823 3100 42.4207 4536

Finally, Output 91.2.2 displays the summary statistics for the second grid point (`East`=75, `North`=75).

Output 91.2.2: Simulation Statistics at Grid Point (`East`=75, `North`=75)

 Simulation Statistics at Selected Grid Points

The UNIVARIATE Procedure
Variable: SVALUE (Simulated Value at Grid Point)

Quantiles (Definition 5)
Level Quantile
100% Max 40.1171
99% 40.1147
95% 40.1131
90% 40.1122
75% Q3 40.1108
50% Median 40.1092
25% Q1 40.1075
10% 40.1062
5% 40.1053
1% 40.1035
0% Min 40.1001

Extreme Observations
Lowest Highest
Value Obs Value Obs
40.1001 7176 40.1167 8980
40.1007 6262 40.1167 9272
40.1011 7383 40.1169 5676
40.1016 7156 40.1170 6514
40.1017 5643 40.1171 5329

For each simulation location, a single realization might result in values that differ significantly from the random field mean at that location. However, the averages of progressively larger numbers of realizations tend to shorten this gap and reduce the simulation variability, as exhibited in the results for the two example locations in Output 91.2.1 and Output 91.2.2. At the limit of an infinite number of realizations, the simulation mean recovers the mean and covariance structure of the random field.