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Ore reserve estimation using localized trend analysis
Mining Science and Technology, 7 (1988) 161-165 Elsevier Science Publishers B.V., Amsterdam - Printed in The Netherlands
161
ORE RESERVE ESTIMATION USING LOCALIZED TREND ANALYSIS E. Alaphia Wright Department of Mining Engineering, University of Zimbabwe, P.O. Box MP 167, Mount Pleasant, Harare (Zimbabwe)
(Received February 18, 1988; accepted March 14, 1988)
ABSTRACT The techniques used in ore reserve estimation vary from the simple method of polygons through heuristics such as distance weighting methods to quite sophisticated kriging procedures of geostatistics. The results from the method of polygons are usually inferior to those of both the heuristics and geostatistics. This paper discusses a heuristic: Localized Trend Analysis (LTA). Most reserve estimating techniques start off by characterizing the deposits as a whole by
means of frequency distributions, general trend surfaces or semi variograms. This is then followed by estimates of grades for smaller mining blocks. L T A starts off by exploiting local grade variation in deriving estimates for smaller mining blocks before proceeding to consider estimates for a deposit as a whole. The derivation of the basic formulae employed in LTA is presented and the estimation procedure is illustrated by means of an example.
INTRODUCTION
matical sophistication. These methods are, in a sense, the heuristics for ore reserve estimation. They inevitably lack the mathematical rigour of geostatistics but will usually give solutions that are clearly superior to those of the method of polygons [1,2]. At the same time, the heuristics share m a n y c o m m o n assumptions with geostatistics and give ore reserve results which do not compare too badly with those of geostatistics [1,2]. Hence in the absence of suitable access to geostatistical expertise and packages, the heuristics present acceptable alternatives for effective ore reserve estimation. The discussions on localized trend analysis (LTA) presented here are based on investigations conducted within on-going research with
Two main extremes can be recognized in the practice of ore reserve estimation. The first centres on the use of simple approaches such as the m e t h o d of polygons and the second involves the various kriging techniques found in geostatistics. The m e t h o d of polygons is clearly inferior to the geostatistical approaches in terms of the qualities of results delivered [1, 2]. The geostatistical approaches on the other hand involve mathematical procedures which are usually not that simple. Between these two extremes are a series of methods such as the distance (or inverse distance) weighting methods and trend surface analysis involving varying degrees of mathe-
162 which the author has been involved. In general the research has as its main aim the development of simple but effective systems for mine evaluation, planning and production control. Such systems are primarily for implementation in a personal computing environment. Development to date includes: algorithms for open pit design and open pit mining sequence planning [3-5]. Continuing work involves ore reserve estimation and truck dispatching. The first part of this presentation discusses the derivation of the basic formulae employed in LTA. The second part demonstrates the use of the LTA procedure by means of an illustration.
TREND PROPERTY AND ORE ESTIMATION Consider three points lying along a trend direction 1-2-3 as shown in Fig. 1. The grades at points 1, 2 and 3 are gl, g2 and g3, respectively. Assuming that the grades gl, g2, and g3 are correlated, then there would be a trend relationship between them. This trend property, which can be expressed as a trend variable, T, is valid only for the relationship connecting the three points and three grades. The trend variable, T, is therefore local and will take different values at different locations and directions within a given deposit. The trend property is such that knowing the distance between points 1 and 2 ( = d12 ) it is possible to estimate g2 from gl and vice versa. The same holds true for g3 from g2. One possible relationship between the grades (gl, g2 and g3), their distances apart (d12 and d23) and the trend variable (T), is: g2 = gl + Td12" and g3 = g2 + from(l)" r -
(1)
Td23-
(2)
g2 - g l d12
(3)
-~
1 X
2
3
X
X
Fig. l. Three points 1, 2 and 3 along a trend direction
Substituting for T from (3) into (2) gives:
g3 = g2 + ( g 2 - 2gl ) "
(4)
Solving for g2 from eqn. (4) gives: g2 =
d12-g3 + d23 " gl dl 3
(5)
Therefore knowing the grades at two positions (1 and 3) along a given trend direction it is possible to estimate the grade at a third position (point 2) lying between the two known points along the same trend direction using eqn. (5). A closer examination of the interpolation model expressed in eqns. (1) and (2) will show that this is none other than the familiar equation for a straight line; y = ax + b. The straight line is in fact the simplest regression model in existence. Estimation procedure The estimation procedure for a given point is made up of several steps: Step 1 Define relevant trend directions passing through the point at which a value is to be estimated. Step 2 Take a defined trend direction and select the two nearest known sample positions having the point of u n k n o w n value between them. Note the value at the two positions concerned. Step 3 Use eqn. (5) to estimate the value at the point in question.
163
Step 4 Repeat steps 2 and 3 for the other trend directions and then do the final estimation of the value at the desired point. Estimated value at a point
The four steps of the estimation procedure will result in a number of estimates for the same point. For instance, with " N " trend directions passing through the point in question, the procedure will result in " N " estimates; namely, E l , E 2 , . . . , E N. The most convenient case is when E 1 -- E 2 =- . . . , = E N ; then the true value, G e , will be: G e = E i ( i = 1, 2 , . . . , N ) , since all the " N " estimates refer to the same point. Unfortunately, this will seldom be the case and so G e will have to be determined by other means. One way of estimating G E is to use the method of least squares. If G e is the true, but unknown, value at the point in question, then we estimate GE such that the function, F, is a minimum, where: N Y'~ ( G E i=1
F=
El) 2
(6)
Differentiating F with respect to G E gives: dF
N
a G e - 2 N G e - 2 Y'~ E i i=1
(7)
Equating d F / d G e to zero and solving for G e gives: 1 N G E = ~ " g E, (8) i=l
Hence G e can be estimated as the arithmetic mean of the various trend estimates. Evidently, the larger the n u m b e r of trend directions, considered, the more accurate the estimate of Ge will be.
dac~ =dcct =dqe =dq,g =200m dbq" =ddc~ =dqf =dqh = 283m # Trend Direcfion ']'
Fig. 2. Sketch for LTA worked example.
been arranged in 4 trend directions as shown by the arrows. It is required to estimate the grade at the location " q " using localized trend analysis. T h e grades shown in the sketch are in % Cu and the relevant distances in metres are also shown. As an illustration of the calculations, extract the details for the trend direction "1" from Fig. 2. We then get the sketch shown in Fig. 3. Now using eqn. (5): E1 = g2 =
with d12 = daq = 200m da3 = dqe = 200m d13 = dae = 4 0 0 m
gl -- ga = 0.719% g3 = ge
Illustration
Figure 2 shows a sketch in which 8 known sample grades (located at a, b , . . . , h ) have
d12 " g3 + d23 " g l dl 3
=
0.638%
We then have (200 × 0.638) + (200 × 0.719) E~ 400 E 1 = 0.679% Cu =
(9)
164
VV-q a
0.719%
q
0.679 % (estimated)
e
0.63B %
Fig. 3. Extract of trend direction 1 for LTA worked example.
Similarly: E 2 = 1.034% Cu E3=
1.350% Cu
E4=
0.634% Cu
F r o m which the mean value for G e is calculated to be: G E = 0.924% Cu N o w at 95% confidence limits we have: Mean G e = 0.924% Cu Lower limit = 0.389% Cu U p p e r limit = 1.459% Cu The use of confidence limits here is not strictly relevant since the various E i, S are "estimates" and not "measurements". However since the estimates refer to the u n k n o w n real value at the same point, the confidence limits (using students t) serve to show the spread of the estimates around the u n k n o w n real value. Comments on the illustration
The illustration was derived from a set of sampled data including a real g r a d e at "q". The real grade at " q " was blanked out and the problem formulated to estimate the grade at "q". The real grade at " q " is 0.915% Cu. As a brief comparison, the estimation of the grade at " q " was also performed using the
inverse of the distance squared method (IDS) of interpolation. The IDS m e t h o d gave a grade at " q " of 0.954% Cu.
D I S C U S S I O N AND C O N C L U S I O N S
Irrespective of the type of trend model employed, four fundamental aspects of ore reserves estimation using localized trend analysis (LTA) are Of importance. (1). The correlation between attribute values is essentially local and as such is valid only within the well-defined local range in question. In this case the range is for the distance 1-2-3. (2). Evidently, the shorter the distances between the points 1, 2 and 3, the more accurate the estimates are likely to be. The accuracies involved will also increase with increasing n u m b e r of trend directions for a given estimate. (3). The estimation procedure is independent of both the frequency distributions of the original sample values and the semi-variogram. As such inferences about the deposit as a whole are to be m a d e from the effects of the individual local estimates. This means that local fluctuations are accounted for prior to the establishment of overall deposit values. Confident limits can be attached to the estimates. (4). The mathematics involved is relatively simple. And, since the computations are based on data from localized ranges, it is not necessary to hold large data files in store. The computations can thus be performed in a personal computing environment.
REFERENCES
1 M.P. Barnes, Comparison of estimation techniques at Sacaton Porphyry Copper Mine, Arizona. In: Computer Assisted Mineral Appraisal and Feasibility, Soc. Min. Eng., AIME, New York, 1980, pp. 123-125.
165 2 H.P. Knudsen, Y.C. Kim and E. Mueller, Comparative study of geostatistical ore reserve estimation methods over the conventional methods. Min. Eng., 30 (1) (1978): 54-58. 3 F.L. Wilke and E.A. Wright, Ermittlung der guenstisten Endauslegung yon Hartgesteinstagebauen mittels Dynamischer Programmierung (Determining the optimal ultimate pit design for hard rock open pit mines using dynamic programming). Erzmetall, 37 (1984): 138-144.
4 F.L. Wilke, K. Mueller and E.A. Wright, Ultimate pit and production scheduling optimization. In: Proc. 18th APCOM Symp. Inst. Min. Metall. London, 1984, pp. 29-38. 5 E.A. Wright, The use of dynamic programming for open pit mine design: some practical implications. Mining Science and Technology, 4 (2) (1987): 97-104.
161
ORE RESERVE ESTIMATION USING LOCALIZED TREND ANALYSIS E. Alaphia Wright Department of Mining Engineering, University of Zimbabwe, P.O. Box MP 167, Mount Pleasant, Harare (Zimbabwe)
(Received February 18, 1988; accepted March 14, 1988)
ABSTRACT The techniques used in ore reserve estimation vary from the simple method of polygons through heuristics such as distance weighting methods to quite sophisticated kriging procedures of geostatistics. The results from the method of polygons are usually inferior to those of both the heuristics and geostatistics. This paper discusses a heuristic: Localized Trend Analysis (LTA). Most reserve estimating techniques start off by characterizing the deposits as a whole by
means of frequency distributions, general trend surfaces or semi variograms. This is then followed by estimates of grades for smaller mining blocks. L T A starts off by exploiting local grade variation in deriving estimates for smaller mining blocks before proceeding to consider estimates for a deposit as a whole. The derivation of the basic formulae employed in LTA is presented and the estimation procedure is illustrated by means of an example.
INTRODUCTION
matical sophistication. These methods are, in a sense, the heuristics for ore reserve estimation. They inevitably lack the mathematical rigour of geostatistics but will usually give solutions that are clearly superior to those of the method of polygons [1,2]. At the same time, the heuristics share m a n y c o m m o n assumptions with geostatistics and give ore reserve results which do not compare too badly with those of geostatistics [1,2]. Hence in the absence of suitable access to geostatistical expertise and packages, the heuristics present acceptable alternatives for effective ore reserve estimation. The discussions on localized trend analysis (LTA) presented here are based on investigations conducted within on-going research with
Two main extremes can be recognized in the practice of ore reserve estimation. The first centres on the use of simple approaches such as the m e t h o d of polygons and the second involves the various kriging techniques found in geostatistics. The m e t h o d of polygons is clearly inferior to the geostatistical approaches in terms of the qualities of results delivered [1, 2]. The geostatistical approaches on the other hand involve mathematical procedures which are usually not that simple. Between these two extremes are a series of methods such as the distance (or inverse distance) weighting methods and trend surface analysis involving varying degrees of mathe-
162 which the author has been involved. In general the research has as its main aim the development of simple but effective systems for mine evaluation, planning and production control. Such systems are primarily for implementation in a personal computing environment. Development to date includes: algorithms for open pit design and open pit mining sequence planning [3-5]. Continuing work involves ore reserve estimation and truck dispatching. The first part of this presentation discusses the derivation of the basic formulae employed in LTA. The second part demonstrates the use of the LTA procedure by means of an illustration.
TREND PROPERTY AND ORE ESTIMATION Consider three points lying along a trend direction 1-2-3 as shown in Fig. 1. The grades at points 1, 2 and 3 are gl, g2 and g3, respectively. Assuming that the grades gl, g2, and g3 are correlated, then there would be a trend relationship between them. This trend property, which can be expressed as a trend variable, T, is valid only for the relationship connecting the three points and three grades. The trend variable, T, is therefore local and will take different values at different locations and directions within a given deposit. The trend property is such that knowing the distance between points 1 and 2 ( = d12 ) it is possible to estimate g2 from gl and vice versa. The same holds true for g3 from g2. One possible relationship between the grades (gl, g2 and g3), their distances apart (d12 and d23) and the trend variable (T), is: g2 = gl + Td12" and g3 = g2 + from(l)" r -
(1)
Td23-
(2)
g2 - g l d12
(3)
-~
1 X
2
3
X
X
Fig. l. Three points 1, 2 and 3 along a trend direction
Substituting for T from (3) into (2) gives:
g3 = g2 + ( g 2 - 2gl ) "
(4)
Solving for g2 from eqn. (4) gives: g2 =
d12-g3 + d23 " gl dl 3
(5)
Therefore knowing the grades at two positions (1 and 3) along a given trend direction it is possible to estimate the grade at a third position (point 2) lying between the two known points along the same trend direction using eqn. (5). A closer examination of the interpolation model expressed in eqns. (1) and (2) will show that this is none other than the familiar equation for a straight line; y = ax + b. The straight line is in fact the simplest regression model in existence. Estimation procedure The estimation procedure for a given point is made up of several steps: Step 1 Define relevant trend directions passing through the point at which a value is to be estimated. Step 2 Take a defined trend direction and select the two nearest known sample positions having the point of u n k n o w n value between them. Note the value at the two positions concerned. Step 3 Use eqn. (5) to estimate the value at the point in question.
163
Step 4 Repeat steps 2 and 3 for the other trend directions and then do the final estimation of the value at the desired point. Estimated value at a point
The four steps of the estimation procedure will result in a number of estimates for the same point. For instance, with " N " trend directions passing through the point in question, the procedure will result in " N " estimates; namely, E l , E 2 , . . . , E N. The most convenient case is when E 1 -- E 2 =- . . . , = E N ; then the true value, G e , will be: G e = E i ( i = 1, 2 , . . . , N ) , since all the " N " estimates refer to the same point. Unfortunately, this will seldom be the case and so G e will have to be determined by other means. One way of estimating G E is to use the method of least squares. If G e is the true, but unknown, value at the point in question, then we estimate GE such that the function, F, is a minimum, where: N Y'~ ( G E i=1
F=
El) 2
(6)
Differentiating F with respect to G E gives: dF
N
a G e - 2 N G e - 2 Y'~ E i i=1
(7)
Equating d F / d G e to zero and solving for G e gives: 1 N G E = ~ " g E, (8) i=l
Hence G e can be estimated as the arithmetic mean of the various trend estimates. Evidently, the larger the n u m b e r of trend directions, considered, the more accurate the estimate of Ge will be.
dac~ =dcct =dqe =dq,g =200m dbq" =ddc~ =dqf =dqh = 283m # Trend Direcfion ']'
Fig. 2. Sketch for LTA worked example.
been arranged in 4 trend directions as shown by the arrows. It is required to estimate the grade at the location " q " using localized trend analysis. T h e grades shown in the sketch are in % Cu and the relevant distances in metres are also shown. As an illustration of the calculations, extract the details for the trend direction "1" from Fig. 2. We then get the sketch shown in Fig. 3. Now using eqn. (5): E1 = g2 =
with d12 = daq = 200m da3 = dqe = 200m d13 = dae = 4 0 0 m
gl -- ga = 0.719% g3 = ge
Illustration
Figure 2 shows a sketch in which 8 known sample grades (located at a, b , . . . , h ) have
d12 " g3 + d23 " g l dl 3
=
0.638%
We then have (200 × 0.638) + (200 × 0.719) E~ 400 E 1 = 0.679% Cu =
(9)
164
VV-q a
0.719%
q
0.679 % (estimated)
e
0.63B %
Fig. 3. Extract of trend direction 1 for LTA worked example.
Similarly: E 2 = 1.034% Cu E3=
1.350% Cu
E4=
0.634% Cu
F r o m which the mean value for G e is calculated to be: G E = 0.924% Cu N o w at 95% confidence limits we have: Mean G e = 0.924% Cu Lower limit = 0.389% Cu U p p e r limit = 1.459% Cu The use of confidence limits here is not strictly relevant since the various E i, S are "estimates" and not "measurements". However since the estimates refer to the u n k n o w n real value at the same point, the confidence limits (using students t) serve to show the spread of the estimates around the u n k n o w n real value. Comments on the illustration
The illustration was derived from a set of sampled data including a real g r a d e at "q". The real grade at " q " was blanked out and the problem formulated to estimate the grade at "q". The real grade at " q " is 0.915% Cu. As a brief comparison, the estimation of the grade at " q " was also performed using the
inverse of the distance squared method (IDS) of interpolation. The IDS m e t h o d gave a grade at " q " of 0.954% Cu.
D I S C U S S I O N AND C O N C L U S I O N S
Irrespective of the type of trend model employed, four fundamental aspects of ore reserves estimation using localized trend analysis (LTA) are Of importance. (1). The correlation between attribute values is essentially local and as such is valid only within the well-defined local range in question. In this case the range is for the distance 1-2-3. (2). Evidently, the shorter the distances between the points 1, 2 and 3, the more accurate the estimates are likely to be. The accuracies involved will also increase with increasing n u m b e r of trend directions for a given estimate. (3). The estimation procedure is independent of both the frequency distributions of the original sample values and the semi-variogram. As such inferences about the deposit as a whole are to be m a d e from the effects of the individual local estimates. This means that local fluctuations are accounted for prior to the establishment of overall deposit values. Confident limits can be attached to the estimates. (4). The mathematics involved is relatively simple. And, since the computations are based on data from localized ranges, it is not necessary to hold large data files in store. The computations can thus be performed in a personal computing environment.
REFERENCES
1 M.P. Barnes, Comparison of estimation techniques at Sacaton Porphyry Copper Mine, Arizona. In: Computer Assisted Mineral Appraisal and Feasibility, Soc. Min. Eng., AIME, New York, 1980, pp. 123-125.
165 2 H.P. Knudsen, Y.C. Kim and E. Mueller, Comparative study of geostatistical ore reserve estimation methods over the conventional methods. Min. Eng., 30 (1) (1978): 54-58. 3 F.L. Wilke and E.A. Wright, Ermittlung der guenstisten Endauslegung yon Hartgesteinstagebauen mittels Dynamischer Programmierung (Determining the optimal ultimate pit design for hard rock open pit mines using dynamic programming). Erzmetall, 37 (1984): 138-144.
4 F.L. Wilke, K. Mueller and E.A. Wright, Ultimate pit and production scheduling optimization. In: Proc. 18th APCOM Symp. Inst. Min. Metall. London, 1984, pp. 29-38. 5 E.A. Wright, The use of dynamic programming for open pit mine design: some practical implications. Mining Science and Technology, 4 (2) (1987): 97-104.