In this section we present suggestions for exercises, no ready solutions for fitting problems.
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1 | 6m | 0.14 | 0.17 |
2 | 12 | 0.20 | 0.26 |
3 | 18 | 0.23 | 0.32 |
4 | 24 | 0.29 | 0.42 |
5 | 30 | 0.33 | 0.42 |
6 | 36 | 0.40 | 0.58 |
7 | 42 | 0.47 | 0.68 |
8 | 48 | 0.48 | 0.54 |
9 | 54 | 0.53 | 0.56 |
10 | 60 | 0.60 | 0.45 |
11 | 66 | 0.65 | 0.60 |
12 | 72 | 0.44 | 0.72 |
13 | 78 | 0.64 | 0.48 |
14 | 84 | 0.54 | 0.40 |
15 | 90 | 0.67 | 0.52 |
16 | 96 | 0.60 | 0.57 |
17 | 102 | 0.72 | 0.44 |
18 | 108 | 0.44 | 0.48 |
19 | 114 | 0.66 | 0.58 |
20 | 120 | 0.45 | 0.64 |
The statistical variance of the samples of the ore body
is
.
- The points
depending on
are presented
in the next diagram
Figure 4.16. Obviously, we have a geometrical anisotropy.
- Parallel to the -axis we draw a straight line at
in respect to
the
-coordinate. This represents the empirical variance measured from the
volumes
in the ore body
, namely
- For each of the two variograms a tangent through the first points towards
the origin
is drawn. It is obvious that
for
is
not equal 0. There is a nugget variance
which has to be determined.
Because of the hypothesis of geometrical anisotropy there can be only one
nugget variance. We then have
- From the crossing points of the tangents with the horizontal line
we find in the Matheron-model 2/3 of the ranges.
We can read:
for the
-direction, which means:
for the
-direction, which means:
- With the relation
the
-variogram may be
geometrically transformed to the
-variogram, such that only an
-variogram must be computed.
- The Matheron-model then is:
- The fitted values for
are to be inserted in the diagram with
the empirical variogram. The easiest way, of course, is to use a computer with
a graphical output device. For the work with pencil and scratch paper we can
use a table for
h/a | 0.000 | 0.001 | 0.002 | 0.003 | 0.004 | 0.005 | 0.006 | 0.007 | 0.008 | 0.009 |
0.000 | 0.000 | 0.001 | 0.003 | 0.004 | 0.006 | 0.007 | 0.009 | 0.010 | 0.012 | 0.013 |
0.010 | 0.015 | 0.016 | 0.018 | 0.019 | 0.021 | 0.022 | 0.024 | 0.025 | 0.027 | 0.028 |
0.020 | 0.030 | 0.031 | 0.033 | 0.034 | 0.036 | 0.037 | 0.039 | 0.040 | 0.042 | 0.043 |
0.030 | 0.045 | 0.046 | 0.048 | 0.049 | 0.051 | 0.052 | 0.054 | 0.055 | 0.057 | 0.058 |
0.040 | 0.060 | 0.061 | 0.063 | 0.064 | 0.066 | 0.067 | 0.069 | 0.070 | 0.072 | 0.073 |
0.050 | 0.075 | 0.076 | 0.078 | 0.079 | 0.081 | 0.082 | 0.084 | 0.085 | 0.087 | 0.088 |
0.060 | 0.090 | 0.091 | 0.093 | 0.094 | 0.096 | 0.097 | 0.099 | 0.100 | 0.102 | 0.103 |
0.070 | 0.105 | 0.106 | 0.108 | 0.109 | 0.111 | 0.112 | 0.114 | 0.115 | 0.117 | 0.118 |
0.080 | 0.120 | 0.121 | 0.123 | 0.124 | 0.126 | 0.127 | 0.129 | 0.130 | 0.132 | 0.133 |
0.090 | 0.135 | 0.136 | 0.138 | 0.139 | 0.141 | 0.142 | 0.144 | 0.145 | 0.147 | 0.148 |
0.100 | 0.150 | 0.151 | 0.152 | 0.154 | 0.155 | 0.157 | 0.158 | 0.160 | 0.161 | 0.163 |
0.110 | 0.164 | 0.166 | 0.167 | 0.169 | 0.170 | 0.172 | 0.173 | 0.175 | 0.176 | 0.178 |
0.120 | 0.179 | 0.181 | 0.182 | 0.184 | 0.185 | 0.187 | 0.188 | 0.189 | 0.191 | 0.192 |
0.130 | 0.194 | 0.195 | 0.197 | 0.198 | 0.200 | 0.201 | 0.203 | 0.204 | 0.206 | 0.207 |
0.140 | 0.209 | 0.210 | 0.212 | 0.213 | 0.215 | 0.216 | 0.217 | 0.219 | 0.220 | 0.222 |
0.150 | 0.223 | 0.225 | 0.226 | 0.228 | 0.229 | 0.231 | 0.232 | 0.234 | 0.235 | 0.236 |
0.160 | 0.238 | 0.239 | 0.241 | 0.242 | 0.244 | 0.245 | 0.247 | 0.248 | 0.250 | 0.251 |
0.170 | 0.253 | 0.254 | 0.255 | 0.257 | 0.258 | 0.260 | 0.261 | 0.263 | 0.264 | 0.266 |
0.180 | 0.267 | 0.269 | 0.270 | 0.271 | 0.273 | 0.274 | 0.276 | 0.277 | 0.279 | 0.280 |
0.190 | 0.282 | 0.283 | 0.284 | 0.286 | 0.287 | 0.289 | 0.290 | 0.292 | 0.293 | 0.295 |
0.200 | 0.296 | 0.297 | 0.299 | 0.300 | 0.302 | 0.303 | 0.305 | 0.306 | 0.308 | 0.309 |
0.210 | 0.310 | 0.312 | 0.313 | 0.315 | 0.316 | 0.318 | 0.319 | 0.320 | 0.322 | 0.323 |
0.220 | 0.325 | 0.326 | 0.328 | 0.329 | 0.330 | 0.332 | 0.333 | 0.335 | 0.336 | 0.337 |
0.230 | 0.339 | 0.340 | 0.342 | 0.343 | 0.345 | 0.346 | 0.347 | 0.349 | 0.350 | 0.352 |
0.240 | 0.353 | 0.355 | 0.356 | 0.357 | 0.359 | 0.360 | 0.362 | 0.363 | 0.364 | 0.366 |
0.250 | 0.367 | 0.369 | 0.370 | 0.371 | 0.373 | 0.374 | 0.376 | 0.377 | 0.378 | 0.380 |
0.260 | 0.381 | 0.383 | 0.384 | 0.385 | 0.387 | 0.388 | 0.390 | 0.391 | 0.392 | 0.394 |
0.270 | 0.395 | 0.397 | 0.398 | 0.399 | 0.401 | 0.402 | 0.403 | 0.405 | 0.406 | 0.408 |
0.280 | 0.409 | 0.410 | 0.412 | 0.413 | 0.415 | 0.416 | 0.417 | 0.419 | 0.420 | 0.421 |
0.290 | 0.423 | 0.424 | 0.426 | 0.427 | 0.428 | 0.430 | 0.431 | 0.432 | 0.434 | 0.435 |
0.300 | 0.437 | 0.438 | 0.439 | 0.441 | 0.442 | 0.443 | 0.445 | 0.446 | 0.447 | 0.449 |
0.310 | 0.450 | 0.451 | 0.453 | 0.454 | 0.456 | 0.457 | 0.458 | 0.460 | 0.461 | 0.462 |
0.320 | 0.464 | 0.465 | 0.466 | 0.468 | 0.469 | 0.470 | 0.472 | 0.473 | 0.474 | 0.476 |
0.330 | 0.477 | 0.478 | 0.480 | 0.481 | 0.482 | 0.484 | 0.485 | 0.486 | 0.488 | 0.489 |
0.340 | 0.490 | 0.492 | 0.493 | 0.494 | 0.496 | 0.497 | 0.498 | 0.500 | 0.501 | 0.502 |
0.350 | 0.504 | 0.505 | 0.506 | 0.508 | 0.509 | 0.510 | 0.511 | 0.513 | 0.514 | 0.515 |
0.360 | 0.517 | 0.518 | 0.519 | 0.521 | 0.522 | 0.523 | 0.524 | 0.526 | 0.527 | 0.528 |
0.370 | 0.530 | 0.531 | 0.532 | 0.534 | 0.535 | 0.536 | 0.537 | 0.539 | 0.540 | 0.541 |
0.380 | 0.543 | 0.544 | 0.545 | 0.546 | 0.548 | 0.549 | 0.550 | 0.552 | 0.553 | 0.554 |
0.390 | 0.555 | 0.557 | 0.558 | 0.559 | 0.560 | 0.562 | 0.563 | 0.564 | 0.565 | 0.567 |
0.400 | 0.568 | 0.569 | 0.571 | 0.572 | 0.573 | 0.574 | 0.576 | 0.577 | 0.578 | 0.579 |
0.410 | 0.581 | 0.582 | 0.583 | 0.584 | 0.586 | 0.587 | 0.588 | 0.589 | 0.590 | 0.592 |
0.420 | 0.593 | 0.594 | 0.595 | 0.597 | 0.598 | 0.599 | 0.600 | 0.602 | 0.603 | 0.604 |
0.430 | 0.605 | 0.606 | 0.608 | 0.609 | 0.610 | 0.611 | 0.613 | 0.614 | 0.615 | 0.616 |
0.440 | 0.617 | 0.619 | 0.620 | 0.621 | 0.622 | 0.623 | 0.625 | 0.626 | 0.627 | 0.628 |
0.450 | 0.629 | 0.631 | 0.632 | 0.633 | 0.634 | 0.635 | 0.637 | 0.638 | 0.639 | 0.640 |
0.460 | 0.641 | 0.643 | 0.644 | 0.645 | 0.646 | 0.647 | 0.648 | 0.650 | 0.651 | 0.652 |
0.470 | 0.653 | 0.654 | 0.655 | 0.657 | 0.658 | 0.659 | 0.660 | 0.661 | 0.662 | 0.664 |
0.480 | 0.665 | 0.666 | 0.667 | 0.668 | 0.669 | 0.670 | 0.672 | 0.673 | 0.674 | 0.675 |
0.490 | 0.676 | 0.677 | 0.678 | 0.680 | 0.681 | 0.682 | 0.683 | 0.684 | 0.685 | 0.686 |
0.500 | 0.687 | 0.689 | 0.690 | 0.691 | 0.692 | 0.693 | 0.694 | 0.695 | 0.696 | 0.698 |
h/a | 0.000 | 0.001 | 0.002 | 0.003 | 0.004 | 0.005 | 0.006 | 0.007 | 0.008 | 0.009 |
0.510 | 0.699 | 0.700 | 0.701 | 0.702 | 0.703 | 0.704 | 0.705 | 0.706 | 0.708 | 0.709 |
0.520 | 0.710 | 0.711 | 0.712 | 0.713 | 0.714 | 0.715 | 0.716 | 0.717 | 0.718 | 0.719 |
0.530 | 0.721 | 0.722 | 0.723 | 0.724 | 0.725 | 0.726 | 0.727 | 0.728 | 0.729 | 0.730 |
0.540 | 0.731 | 0.732 | 0.733 | 0.734 | 0.736 | 0.737 | 0.738 | 0.739 | 0.740 | 0.741 |
0.550 | 0.742 | 0.743 | 0.744 | 0.745 | 0.746 | 0.747 | 0.748 | 0.749 | 0.750 | 0.751 |
0.560 | 0.752 | 0.753 | 0.754 | 0.755 | 0.756 | 0.757 | 0.758 | 0.759 | 0.760 | 0.761 |
0.570 | 0.762 | 0.763 | 0.764 | 0.765 | 0.766 | 0.767 | 0.768 | 0.769 | 0.770 | 0.771 |
0.580 | 0.772 | 0.773 | 0.774 | 0.775 | 0.776 | 0.777 | 0.778 | 0.779 | 0.780 | 0.781 |
0.590 | 0.782 | 0.783 | 0.784 | 0.785 | 0.786 | 0.787 | 0.788 | 0.789 | 0.790 | 0.791 |
0.600 | 0.792 | 0.793 | 0.794 | 0.795 | 0.796 | 0.797 | 0.798 | 0.799 | 0.800 | 0.801 |
0.610 | 0.802 | 0.802 | 0.803 | 0.804 | 0.805 | 0.806 | 0.807 | 0.808 | 0.809 | 0.810 |
0.620 | 0.811 | 0.812 | 0.813 | 0.814 | 0.815 | 0.815 | 0.816 | 0.817 | 0.818 | 0.819 |
0.630 | 0.820 | 0.821 | 0.822 | 0.823 | 0.824 | 0.824 | 0.825 | 0.826 | 0.827 | 0.828 |
0.640 | 0.829 | 0.830 | 0.831 | 0.832 | 0.832 | 0.833 | 0.834 | 0.835 | 0.836 | 0.837 |
0.650 | 0.838 | 0.839 | 0.839 | 0.840 | 0.841 | 0.842 | 0.843 | 0.844 | 0.845 | 0.845 |
0.660 | 0.846 | 0.847 | 0.848 | 0.849 | 0.850 | 0.850 | 0.851 | 0.852 | 0.853 | 0.854 |
0.670 | 0.855 | 0.855 | 0.856 | 0.857 | 0.858 | 0.859 | 0.860 | 0.860 | 0.861 | 0.862 |
0.680 | 0.863 | 0.864 | 0.864 | 0.865 | 0.866 | 0.867 | 0.868 | 0.868 | 0.869 | 0.870 |
0.690 | 0.871 | 0.872 | 0.872 | 0.873 | 0.874 | 0.875 | 0.875 | 0.876 | 0.877 | 0.878 |
0.700 | 0.879 | 0.879 | 0.880 | 0.881 | 0.882 | 0.882 | 0.883 | 0.884 | 0.885 | 0.885 |
0.710 | 0.886 | 0.887 | 0.888 | 0.888 | 0.889 | 0.890 | 0.890 | 0.891 | 0.892 | 0.893 |
0.720 | 0.893 | 0.894 | 0.895 | 0.896 | 0.896 | 0.897 | 0.898 | 0.898 | 0.899 | 0.900 |
0.730 | 0.900 | 0.901 | 0.902 | 0.903 | 0.903 | 0.904 | 0.905 | 0.905 | 0.906 | 0.907 |
0.740 | 0.907 | 0.908 | 0.909 | 0.909 | 0.910 | 0.911 | 0.911 | 0.912 | 0.913 | 0.913 |
0.750 | 0.914 | 0.915 | 0.915 | 0.916 | 0.917 | 0.917 | 0.918 | 0.919 | 0.919 | 0.920 |
0.760 | 0.921 | 0.921 | 0.922 | 0.922 | 0.923 | 0.924 | 0.924 | 0.925 | 0.926 | 0.926 |
0.770 | 0.927 | 0.927 | 0.928 | 0.929 | 0.929 | 0.930 | 0.930 | 0.931 | 0.932 | 0.932 |
0.780 | 0.933 | 0.933 | 0.934 | 0.934 | 0.935 | 0.936 | 0.936 | 0.937 | 0.937 | 0.938 |
0.790 | 0.938 | 0.939 | 0.940 | 0.940 | 0.941 | 0.941 | 0.942 | 0.942 | 0.943 | 0.943 |
0.800 | 0.944 | 0.945 | 0.945 | 0.946 | 0.946 | 0.947 | 0.947 | 0.948 | 0.948 | 0.949 |
0.810 | 0.949 | 0.950 | 0.950 | 0.951 | 0.951 | 0.952 | 0.952 | 0.953 | 0.953 | 0.954 |
0.820 | 0.954 | 0.955 | 0.955 | 0.956 | 0.956 | 0.957 | 0.957 | 0.958 | 0.958 | 0.959 |
0.830 | 0.959 | 0.960 | 0.960 | 0.960 | 0.961 | 0.961 | 0.962 | 0.962 | 0.963 | 0.963 |
0.840 | 0.964 | 0.964 | 0.965 | 0.965 | 0.965 | 0.966 | 0.966 | 0.967 | 0.967 | 0.968 |
0.850 | 0.968 | 0.968 | 0.969 | 0.969 | 0.970 | 0.970 | 0.970 | 0.971 | 0.971 | 0.972 |
0.860 | 0.972 | 0.972 | 0.973 | 0.973 | 0.974 | 0.974 | 0.974 | 0.975 | 0.975 | 0.975 |
0.870 | 0.976 | 0.976 | 0.976 | 0.977 | 0.977 | 0.978 | 0.978 | 0.978 | 0.979 | 0.979 |
0.880 | 0.979 | 0.980 | 0.980 | 0.980 | 0.981 | 0.981 | 0.981 | 0.982 | 0.982 | 0.982 |
0.890 | 0.983 | 0.983 | 0.983 | 0.983 | 0.984 | 0.984 | 0.984 | 0.985 | 0.985 | 0.985 |
0.900 | 0.986 | 0.986 | 0.986 | 0.986 | 0.987 | 0.987 | 0.987 | 0.987 | 0.988 | 0.988 |
0.910 | 0.988 | 0.988 | 0.989 | 0.989 | 0.989 | 0.989 | 0.990 | 0.990 | 0.990 | 0.990 |
0.920 | 0.991 | 0.991 | 0.991 | 0.991 | 0.992 | 0.992 | 0.992 | 0.992 | 0.992 | 0.993 |
0.930 | 0.993 | 0.993 | 0.993 | 0.993 | 0.994 | 0.994 | 0.994 | 0.994 | 0.994 | 0.995 |
0.940 | 0.995 | 0.995 | 0.995 | 0.995 | 0.995 | 0.996 | 0.996 | 0.996 | 0.996 | 0.996 |
0.950 | 0.996 | 0.996 | 0.997 | 0.997 | 0.997 | 0.997 | 0.997 | 0.997 | 0.997 | 0.998 |
0.960 | 0.998 | 0.998 | 0.998 | 0.998 | 0.998 | 0.998 | 0.998 | 0.998 | 0.998 | 0.999 |
0.970 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 | 0.999 |
0.980 | 0.999 | 0.999 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
0.990 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 | 1.000 |
It is to determine: and the theoretical variogram model. The
fitting is left to the reader.
Distance | Empirical | Number |
between the | Variogram | of |
Samples | Pairs | |
2 | 0.74 | 1222 |
4 | 1.10 | 1194 |
6 | 1.34 | 1186 |
8 | 1.58 | 1152 |
10 | 1.72 | 1137 |
12 | 1.81 | 1120 |
14 | 1.87 | 1095 |
16 | 1.90 | 1077 |
18 | 1.93 | 1055 |
20 | 1.92 | 1026 |
22 | 1.95 | 1011 |
24 | 2.01 | 990 |
26 | 2.09 | 969 |
28 | 2.16 | 950 |
30 | 2.25 | 919 |
32 | 2.29 | 899 |
34 | 2.38 | 886 |
36 | 2.35 | 860 |
38 | 2.36 | 848 |
40 | 2.39 | 825 |
42 | 2.48 | 814 |
44 | 2.52 | 787 |
46 | 2.56 | 779 |
48 | 2.55 | 767 |
50 | 2.49 | 750 |
52 | 2.59 | 736 |
54 | 2.61 | 722 |
56 | 2.64 | 705 |
58 | 2.68 | 689 |
60 | 2.62 | 675 |
62 | 2.52 | 657 |
64 | 2.59 | 639 |
66 | 2.53 | 628 |
68 | 2.47 | 612 |
70 | 2.56 | 597 |
It should be determined if we have the case of a simple or a nested spherical model.
The data should be graphically presented, the sills and the ranges estimated and the variogram model computed and compared with the empirical values.
- The given variogram data are presented in the next Figure 4.17.
Because of the obvious two bends we decide for a twice nested variogram model
-
- A straight line through the first two points of the empirical
variogram which crosses the ordinate at 0.4 lets us decide for .
-
- A horizonal line through the highest, more or less stable part
of the variogram yields to the total sill
-
- The approximate tangent of , moved parallel
to
, shows the crossing point with the ordinate at
-
- The part of on the total variogram subtracted from
the tangent
, gives the crossing point of the line
the value
and therefore
With the so found parameters the first trial of the fitting is finished:
50 m.
Now one should compute the fitted variogram, draw it in the diagram and modify
the parameters if necessary.