FPG (FIELD PARAMETRIC GEOSTATISTICS)

Geostatistical rigorous solutions for highly-skewed distributions

Machado R. S., Costa J. F. C. L. and Koppe J. C., UFRGS, Brazil; Armony M., MAIM, Brazil

1. The variography and the kriging procedures do not use all the available information. Part of it is abandoned.

2. There are two physical axioms that must be added to ore reserve estimation and one of them is not taken into account in the classical Geostatistical theory.

3. It is usual to work in Geostatistics assuming that the samples are punctual and the precisions of the grade measurements are absolute.

Classical Geostatistics is not able to produce reliable results for highly skewed distributions in Ore Reserve Estimation due to three reasons:

The mineral resource analyst adjust the high-grade 90 to a lower value. Indeed he limits the sample extension.

The abandoned information and a priori wrong inference:

• 1

• 2 • 3

• 90

Using the discrete spectrum

Possible representation for cumulative frequency of the grades (pure Stieltjes approach: a sequence of points).

The conditional probability of picking a sample with grade 90 is 1/87 of picking a

sample with grade 2 or 3

The two underlying axioms

1. Data representativeness: a sample represents more than itself

2. Grade continuity

The theory is based upon two underlying axioms that can be clearly observed when analysing the way macro phenomena variables are evaluated.

Polygons

Examples

Examples Sections

Examples Cells

• Grade and Extension of the sample.

• The extension surrounds the sample (continuity).

• Classical Geostatistics uses only the continuity (Random Function).

• Each sample must be represented by a field, like an electron: the grade and the extension.

We need a variable that represents both the grade and the extension.

All former methods use two variables

Created from the original distribution of grades: the extension. This variable may have its creation conditioned by the declustering weights and sometimes by density. It can represent an area, a volume, a tonnage etc.

Extension of a sample Extension of a grade: sum of all sample extensions with the same grade.

Building a new variable

The cumulative values of grades extensions. This kind of variable is called a global variable and carry both information: grade and extension of each grade. There is an isomorphism between grade and its cumulative extension. Field parametric Geostatistics is the theory that proves that the cumulative extension is the variable to be used.

Extension function:

The curve of the cumulative extensions is not: • a simple Uniform Transform. • a Probability Function. • a Distribution Function.

Samples variogram

Variograms for the synthetic 140 data of GSLib.

• FPG transforms bad or noisy variograms into well behaved variograms

Examples

• FPG reduces automatically the weight of the high grades without manual intervention by using the extension function

FPG variogram

Variograms for the synthetic 140 data of GSLib.

• FPG transforms bad or noisy variograms into well behaved variograms

Examples

• FPG reduces automatically the weight of the high grades without manual intervention by using the extension function

•Data analysis

•Declustering (if necessary)

• FPG transform

•Variography

• Kriging

•Back-transform

If the distribution is not highly skewed, Classical Geostatistics and FPG kriging yield similar results.

Procedure guidelines

• Data analysis and finding inflexion point

• Lower grades: use Classical Statistics

• Higher grades: use FPG

• For both: variography and kriging

• Back-transform for FPG results

• Indicator variogram and kriging

• The final grade of each block is obtained by weighting, using the proportions given by indicator kriging.

Procedure guidelines for two populations

Gold deposit data in Amapa

Example of two populations

V variable 470 samples

U variable 275 samples

Case study: Walker Lake

REFERENCE : 78,000 samples

V variable 470 samples

U variable 275 samples

Case study: Walker Lake

REFERENCE : 78,000 samples

VARIABLE Count Minimum Maximum Mean Std. Dev.

Variance Coef. of

Variation

U 275 0 5190.1 604.08 766.01 586769.89 1.27

U (declust.) 275 0 5190.1 555.00 684.79 468932.61 1.23

V 470 0 1528.1 435.30 299.56 89738.06 0.69

V (declust.) 470 0 1528.1 408.53 293.23 85985.06 0.72

Table 1 – Statistical parameters for U and V variables

VARIABLE Count Min. Max. Mean Std. Dev.

Variance Coef. of

Variation

U (Reference) 3120 0.00 2844.55 266.04 358.81 128744.62 1.35

U (OK) 3101 52.51 2500.58 542.88 314.26 98759.35 0.58

U (cap. Q97) 3120 61.86 1468.65 543.20 260.68 67954.06 0.48

U (E-type 1.5) 3120 7.20 3058.02 464.56 418.14 174841.06 0.90

U (E-type 3.0) 3120 7.20 2316.51 428.82 343.55 118026.60 0.80

U (FPG) 3042 4.84 1590.44 255.22 225.40 50805.16 0.88

Table 2 – Statistical parameters for the U estimates

VARIABLE Count Min. Max. Mean Std. Dev.

Variance Coef. of

Variation

V (Reference) 3120 0.00 1378.12 277.98 228.66 52285.40 0.82

V (OK) 3120 3.81 1052.34 300.44 171.73 29491.19 0.57

V (cap. Q90) 3120 8.70 814.56 304.92 153.93 23694.44 0.50

V (E-type 1.5) 3120 20.74 1295.85 302.10 217.77 47423.77 0.72

V (E-type 3.0) 3120 20.74 1125.01 296.64 202.58 41038.66 0.68

V (FPG) 3120 2.26 922.68 261.84 181.45 32924.10 0.69

Table 3 – Statistical parameters for the V estimates

For U and V, FPG provided better results both globally (Table 2 and 3) and locally.

Conclusions

Software options

The authors thank CNPq and UFRGS for their support. MACHADO R. S. Mining Engineer, MEng Candidate, Federal University of Rio Grande do Sul (UFRGS) Email: rochana.machado@gmail.com

ARMONY M. Geostatistician and Physicist, MSc, PhD, MAIM Ltda. Email: miguelarmony@gmail.com

COSTA J. F. C. L. Mining Engineer, MSc, PhD, Federal University of Rio Grande do Sul (UFRGS) Email: jfelipe@ufrgs.br

KOPPE J. C. Mining Engineer, MSc, DSc, Federal University of Rio Grande do Sul (UFRGS) Email: jkoppe@ufrgs.br