Fitting of a Spherical Variogram Model

In this section we present suggestions for exercises, no ready solutions for fitting problems.

(i)

Samples $(v)$ of an ore body $(V)$ of size $750/ m$ in direction $EW$ and $330/ m$ in $NS$ on a grid of $6 m$ are available. Because the thickness of the deposit varies, the accumulation of the analyses in $[/%.m]$ (grade times thickness) is considered. For this data the variogram $/hat{/gamma}(/boma{h})$ has been calculated (see Table 4.5).

Table 4.5: Variogram Data: Accumulation.

	$h$	$/hat{/gamma}({/boma h}) EW$	$/hat{/gamma}({/boma h}) NS$
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 $(v)$ of the ore body $(V)$ is $/sigma^{2}(v/V)=0.55 [m./%]^{2}$.

- The points $/gamma({/boma h})$ depending on ${/boma h}$ are presented in the next diagram Figure 4.16. Obviously, we have a geometrical anisotropy.

Figure 4.16: Variogram Fitting on Accumulation Data.

- Parallel to the $h$-axis we draw a straight line at $0.55$ in respect to the $y$-coordinate. This represents the empirical variance measured from the volumes $v$ in the ore body $V$, namely

- For each of the two variograms a tangent through the first points towards the origin $({/boma h}={/boma 0})$ is drawn. It is obvious that $/gamma({/boma h})$ for ${/boma h} /rightarrow {/boma 0}$ is not equal 0. There is a nugget variance $C_{o}$ which has to be determined. Because of the hypothesis of geometrical anisotropy there can be only one nugget variance. We then have $C_{1}=C-C_{o}=.55 - .09=.46.$

- From the crossing points of the tangents with the horizontal line $/gamma(/infty)=0.55$ we find in the Matheron-model 2/3 of the ranges. We can read:

$2/3 a_{1}=55$ for the $EW$-direction, which means: $a_{1}=82.5=a.$
$2/3 a_{2}=37$ for the $NS$-direction, which means: $a_{2}=55.5.$

- With the relation $q=a_{1}/a_{2}=1.49$ the $NS$-variogram may be geometrically transformed to the $EW$-variogram, such that only an $isotropical$ $EW$-variogram must be computed.

- The Matheron-model then is:

- The fitted values for $/gamma({/boma h})$ 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

where the values are easy to be transformed. Table 4.6 shows such values of $/gamma(/frac{h}{a})$ for the spherical model with 3 digits after the comma, which is sufficient for practical purposes.

Table: Values of $/gamma (/frac{h}{a}) = /frac{3}{2} (/frac{h}{a}) - /frac{1}{2} (/frac{h}{a})^3$.

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

Table 4.6: Continuation.

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

(ii)

Values of an empirical variogram of a deposit of porphyry-molybden are given, the variance of the samples is .81, see Table 4.7.

Table 4.7: Variogram Data of a Deposit of Porphyry-molybden.

$h$	$/hat{/gamma}(h)$
200'	0.43
282'	0.57
400'	0.63
488'	0.75
564'	0.85
600'	0.85
800'	0.87
1000'	0.88
1200'	0.87
1400'	0.85
1600'	0.80

It is to determine: $C_{o}, C, a$ and the theoretical variogram model. The fitting is left to the reader.

(iii)

Let be given some data of a ore deposit of nickel, see Table 4.8.

Table 4.8: Variogram Data of a Nickel Deposit.

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

Figure 4.17: Fitting of 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 $C_{o}=0.4$.

-

- A horizonal line through the highest, more or less stable part of the variogram yields to the total sill

The range of $/gamma_{2}$ is between 46 und 54 $m$ and we choose $a_{2}=50 m$, i.e. $2/3a_{2}=33.3 m$, which fixes the point

.

-

- The approximate tangent of $/gamma_{2}$, moved parallel to

, shows the crossing point with the ordinate at

It follows that $C_{2}=C - C_{o} - C_{1}= 2.55 - 1.4=1.15$ and $C_{1}=1.4 - 0.4=1.0.$

-

- The part of $/gamma_{2}$ on the total variogram subtracted from the tangent

, gives the crossing point of the line $C_{o} + C_{1}$ the value $/frac{2}{3} a_{1}=7 m$ and therefore $a_{1}=10.5 m.$ With the so found parameters the first trial of the fitting is finished:
$C_{o}=0.4, C_{1}=1.0, C_{2}=1.15, a_{1}=10.5 m, a_{2} =$ 50 m.
Now one should compute the fitted variogram, draw it in the diagram and modify the parameters if necessary.