pilot point optimization of mining boundaries for...

Original Paper

Pilot Point Optimization of Mining Boundaries for LateriticMetal Deposits: Finding the Trade-off Between Dilutionand Ore Loss

Yasin Dagasan ,1,3 Philippe Renard,2 Julien Straubhaar,2 Oktay Erten,1 and Erkan Topal1

Received 24 January 2018; accepted 19 April 2018Published online: 28 April 2018

Geological contacts in lateritic metal deposits (footwall topographies) often delineate theorebody boundaries. Spatial variations seen in such contacts are frequently higher than thosefor the metal grades of the deposit. Therefore, borehole spacing chosen based on the gradevariations cannot adequately capture the geological contact variability. Consequently,models created using such boreholes cause high volumetric uncertainties in the actual andtargeted ore extraction volumes, which, in turn, lead to high unplanned dilution and orelosses. In this paper, a method to design optimum ore/mining boundaries for lateritic metaldeposits is presented. The proposed approach minimizes the dilution/ore losses and com-prises two main steps. First, the uncertainty on the orebody boundary is represented using aset of stochastic realizations generated with a multiple-point statistics algorithm. Then, theoptimal orebody boundary is determined using an optimization technique inspired by amodel calibration method called Pilot Points. The pilot points represent synthetic elevationvalues, and they are used to construct smooth mining boundaries using the multilevel B-spline technique. The performance of a generated surface is evaluated using the expectedsum of losses in each of the stochastic realizations. The simulated annealing algorithm isused to iteratively determine the pilot point values which minimize the expected losses. Theresults show a significant reduction in the dilution volume as compared to those obtainedfrom the actual mining operation.

KEY WORDS: Multiple-point statistics, Direct sampling, Bauxite mining, Laterite simulated anneal-ing, Optimization, Dig limit.

INTRODUCTION

Given a laterite-type bauxite deposit formedfrom tropically weathered mafic–ultramafic com-plexes, the bauxite mineral exists in the soil horizons

(Erten 2012). Therefore, the deposit can be minedeasily by a front-end loader due to the free-flowcharacteristics of the loose soil. Being an underlyinggeological unit, ferricrete is very likely to dilute thebauxite ore during mining operations due to poorlydefined geological interface between bauxite andferricrete units. Although this dilution can partly bealleviated by the front-end loader operator, whosubjectively discriminates the bauxite ore from theferricrete based on the hardness and color differ-ences of the geological units at the time of mining, itstill cannot be avoided entirely.

1Department of Mining and Metallurgical Engineering, Western

Australian School of Mines Curtin University, Kalgoorlie, WA

6430, Australia.2Centre for Hydrogeology and Geothermics, University of Neu-

chatel, rue Emile-Argand 11, 2000 Neuchatel, Switzerland.3To whom correspondence should be addressed; e-mail:

[email protected]

153

1520-7439/19/0100-0153/0 � 2018 International Association for Mathematical Geosciences

Natural Resources Research, Vol. 28, No. 1, January 2019 (� 2018)

https://doi.org/10.1007/s11053-018-9380-9

http://orcid.org/0000-0002-5765-8773

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Ferricrete dilution is the major cause of highsilica content in bauxite ore, as it is tremendouslyrich in silica-bearing minerals such as kaolinite andquartz (Morgan 1995). The contact topography be-tween the bauxite and ferricrete units is ratherundulating and cannot be modeled satisfactorily byusing an economically viable drilling program, as thedrill spacing is usually determined based on thecontinuity and variation in the aluminum grade(Hartman and Mutmansky 2002; Singh 2007). Inother words, because the peaks and troughs cannotbe sampled adequately, they cannot be inferredfrom the geostatistical estimates either (Philip andWatson 1986). This situation is illustrated in Fig-ure 1. Failing to model the contact surface accu-rately introduces a major uncertainty, which maythen lead to: (1) inaccurate calculations of the orevolume/tonnage and the quantity of the caustic sodabeing consumed; and (2) subjective ore extractionstrategies by the front-end loader operators.

There are several ways to reduce the uncer-tainty in the contact surface and its possible conse-quences. One of the easiest ones would be toconduct a dense drilling program to capture thepeaks and troughs of the contact surface. However,this would dramatically increase the associated costsmaking the operation less profitable and even notfeasible at all. Another way to reduce the uncer-tainty in the contact surface models is the use of

geophysical methods to contribute to the orebodydelineation (Campbell 1994; Fallon et al. 1997).Among the geophysical methods, ground-penetrat-ing radar (GPR) has been efficiently used to im-prove the delineation of ore/waste boundaries inlateritic ore bodies (Francke and Nobes 2000;Francke and Parkinson 2000; Francke and Yelf 2003;Francke and Utsi 2009; Francke 2010, 2012a, b;Barsottelli-Botelho and Luiz 2011; Dagasan et al.2018). However, GPR surveys alone cannot replacethe traditional drilling due to their lack of accuracy.They are most efficiently used as secondary infor-mation to complement borehole data through geo-statistical data integration techniques (Erten 2012).Applications of such data integration presented byErten (2012) and Erten et al. (2013, 2015) demon-strate the benefits of using secondary information onmodel precision. However, even though a bettermodel representing the ore/waste contact surface isattained, the large spatial variations inherent to theore/waste interface limit the mining equipment totrack down a given contact surface accurately.

Due to large spatial variations and the uncer-tainties inherent to the geological contact, anyexcavation surface inevitably causes dilution and orelosses. Although this problem shows similarity todig-limit problems in open-pit mining, which hasbeen covered by several studies such as Norrena andDeutsch (2001, 2002), Richmond (2004), Richmond

Pisoli�c Bauxite

Ferricrete

Top Soil

Es�mated surface

Boreholes

Actual surface

Dilu�on Ore LossOre Zone

Figure 1. The peaks and troughs of the actual ore/waste interface cannot be detected by an economically viable

drilling spacing. This results in an inaccurate estimation of the ore/waste contact (after Erten 2012).

154 Dagasan, Renard, Straubhaar, Erten, and Topal

and Beasley (2004), Isaaks et al. (2014), Ruisecoet al. (2016), Ruiseco and Kumral (2017) and Sariand Kumral (2018), the problem with lateritic de-posits is rather specific due to the nature of free-digging mining method. Research on finding theoptimum elevation values for a lateritic nickel minehas been carried out by McLennan et al. (2006), butthe focus was to optimize the dilution and ore losses.The approach did not put strong emphasis onequipment selectivity due to low dilution/ore lossratio and good equipment selectivity.

The aim of this research is to design optimumextraction boundaries for lateritic metal depositsbased on a simulated ore/waste interface. Theproposed approach can be used to generate miningboundaries minimizing the unplanned dilution andore loss as well as increasing the mining equip-ment flexibility. It is inspired by a model calibra-tion technique, which is frequently used inhydrogeology, called pilot points. In this technique,several pilot points are first placed in the area tobe mined out. These pilot points represent syn-thetic elevation values and act as points control-ling the shape of proposed extraction boundaries.The elevation values at the pilot points are itera-tively modified in order to find a smooth excava-tion surface minimizing the possible dilution andore losses. Multilevel B-spline method (MBS) (Leeet al. 1997) was used to create a smooth surface byinterpolating the values at the pilot point locationswith a predefined smoothness parameter. Thelosses associated with a decision surface are cal-culated using several hundreds of equiprobablerealizations generated using the direct sampling(DS) (Mariethoz et al. 2010) multiple-point statis-tics algorithm. This makes the generated excava-tion surfaces account for the uncertainties in theore/waste interface. The elevation values at thepilot point locations were iteratively optimizedusing the simulated annealing (SA) algorithm(Kirkpatrick et al. 1983), which uses the sum ofthe losses in all the realizations as the objectivefunction. The pilot point values yielding minimumlosses were then employed to construct the sug-gested extraction surface.

REVIEW OF UNDERLYING METHODS

The following subsections provide the requiredbackground information to comprehend these

methods, which form the foundations of themethodology described in section ‘‘Methodology.’’

The Direct Sampling MPS Algorithm

The direct sampling (DS) is a pixel-based MPSalgorithm used to simulate a random function Z(x)on a simulation grid (SG) (Mariethoz et al. 2010). Itstochastically reproduces the spatial or temporalpatterns in the simulation domain by integrating thedatasets from analogue sites through training images(TI) (Oriani et al. 2014; Pirot et al. 2014). A TIserves as a conceptual geological model and containsspatial structures that are thought to exist in thesimulation area (Guardiano and Srivastava 1992).The DS uses the spatial patterns in the TI tostochastically simulate a random function Z(x). Thesteps to perform the simulations are as follows:

1. Migrate any available conditioning data tothe SG.

2. Visit a non-informed grid node at x followinga predefined random or regular path.

3. Determine n number of closest informednodes at fx1; x2; . . . ; xng.

4. Define the lag vectors L ¼ fh1; h2; . . . ; hng,where hi ¼ xi � x, to construct the data eventdnðx;LÞ ¼ fZðxþ h1Þ; . . . ;Zðxþ hnÞg.

5. Randomly scan the TI at y locations andcalculate the distance between dnðx;LÞ anddnðy;LÞ ¼ fZðyþ h1Þ; . . . ;Zðyþ hnÞg until itfalls below a threshold t or a maximum scanfraction f is reached.

6. Take the pattern as the best match and pastethe central node Z(y) to the grid node at xlocation.

7. Repeat the steps 2–6 until all the grid nodesare informed.

The DS algorithm also makes it possible formultivariate simulations of m variables, whichare spatially dependent by an unknown function(Mariethoz et al. 2010). This is basically carriedout by computing the distances between thejoint data events dnðxÞ and dnðyÞ of m variablesin both the SG and the TI, respectively. In thisresearch, the MPS simulations were carried outby calling the DS algorithm, which is coded inC, from R software (Team 2017). The version ofthe DS used is called DeeSse (Straubhaar 2016).Detailed information on the algorithm can be

155Pilot Point Optimization of Mining Boundaries for Lateritic Metal Deposits

found in Mariethoz et al. (2010), Meerschmanet al. (2013) and Straubhaar (2016).

Pilot Point Method (PPM)

The PPM is an inverse modeling technique thatis commonly used to calibrate groundwater models(Jung 2008). It was first suggested by de Marsilyet al. (1984) and later modified by several re-searchers (Certes and de Marsily 1991; LaVenueet al. 1995; RamaRao et al. 1995; Oliver et al. 1996;Cooley 2000; Alcolea et al. 2006). The primarymotivation of the PPM is to overcome the non-uniqueness and instability problems of the previousinverse techniques using a reduced parameter space.In this method, several calibration points are firstchosen from the model domain where there are noconductivity measurements taken. These points arecalled pilot points and represent synthetic conduc-tivity values to be iteratively calibrated by mini-mizing the squared errors between the actual andobserved head values. At every step, the pilot pointvalues are used to generate the conductivity fieldusing a geostatistical interpolation technique with aparticular prescribed spatial structure inferred fromthe measurements.

In this research, the PPM was tailored to amining application. Rather than calibrating theconductivity field, the method was used to createoptimum mining boundaries. The pilot points lo-cated within the modeling domain control the shapeof the mining boundaries and were iterated to seekthe pilot point values yielding minimized dilutionand ore losses. The details of our proposed miningapplication are explained in the following sections.

Multilevel B-Splines

The MBS method was used to interpolate orapproximate a scattered dataset (Lee et al. 1997).Given a scattered dataset P ¼ xc; yc; zcð Þf g on adomain, the method uses zc values at xc; ycð Þ loca-tions to carry out the approximations. A functionf x; yð Þ approximating the values zc at xc; ycð Þ loca-tions was sought to interpolate the Z field. To per-form this, the method utilizes a hierarchy of controllattices U0;U1; . . . ;Uh overlain the domain . Each ofthe control lattices Uk contains a different number ofcontrol points with varying spacing. The spacingbetween the control points of a Uk is always halved

for the subsequent control lattice Ukþ1. Therefore,the initial control lattice U0 becomes the coarsestand Uh as the finest. Approximation with thecoarsest control lattice U0 comprises the first step ofthe MBS method yielding f0 function. Being an ini-tial smooth approximation, f0 results in a deviation

D1zc ¼ zc � f0ðxc; ycÞ for each point xc; yc; zcð Þ. Thealgorithm proceeds by using the next control latticeU1 to generate a function f1 which approximates the

preceding deviation P1 ¼ xc; yc;D1zc

� �� . A better

approximation with less departure from the originaldata points P would then be obtained by the sum off0 þ f1. This would result in the deviation

D2zc ¼ zc � f0ðxc; ycÞ � f1ðxc; ycÞ. Therefore, thedeviation for a level k can be calculated as

Dkzc ¼ zc �Pk�1

i¼0 fiðxc; ycÞ. Since the origin of theapproach creates a surface approximating the pointsP, the interpolation is achieved through a sufficientlysmall finest control lattice Uh. The introduction ofthe adaptive control lattice hierarchy helps toachieve finer lattices with a reasonable memoryrequirement. More information regarding the theorycan be found in Lee et al. (1997). The MBS methodin this research was implemented using the MBApackage created for the R statistics software (Finleyand Banerjee 2010).

Simulated Annealing (SA) Algorithm

Simulated annealing (SA) is one of thestochastic optimization techniques for solving globaloptimization problems (Kirkpatrick et al. 1983;Xiang et al. 2013). The method finds the globalminimum of an objective function by mimicking theannealing process of a molten metal. The artificialtemperatures used in the algorithm allows to regu-late the cooling schedule and to introduce stochas-ticity. This stochasticity is basically used to avoid thesolution from trapping inside a local minimum bychanging the probability of acceptance throughoutthe cooling schedule.

Given an objective function f(x) with the deci-sion variables x ¼ ðx1; x2; . . . ; xnÞ, the SA algorithmutilizes the following to attain a global minimum(Sun and Sun 2015):

1. Set a high initial temperature value T0 andan initial solution x0 to evaluate the objectivefunction E0 ¼ f ðx0Þ.

2. Propose a new candidate solution xiþ1:


� Propose a candidate solution xiþ1 based onthe current one ( xi) through a predefinedvisiting distribution.

� Evaluate the energy difference DE ¼f ðxiþ1Þ � f ðxiÞ to observe the change in theobjective function for the candidate solution.

� Accept the iteration if the candidate solutionreduces the objective function, DE\0.

� If the new candidate yields a greater objectivefunction value, accept or reject the solutionbased on a probability of acceptance criterion.

3. Repeat step 2 for L number of iterationsholding T constant.

4. Reduce the temperature to Tnþ1 based on acooling function.

5. Repeat steps 2–4 until the convergence isachieved.

In this research, generalized simulated anneal-ing (GSA) method (Tsallis and Stariolo 1996)was used to optimize the pilot point values. Itmakes use of the distorted Cauchy–Lorentzvisiting distribution to seek for an optimumsolution (Tsallis and Stariolo 1996). The GSAoffers different options for the stopping criteriasuch as maximum running time, maximumfunction calls, maximum iteration number or athreshold value for the objective function. Theimplementation of the GSA was performedusing the GenSA package of the R statisticssoftware (Xiang et al. 2013). The default SAparameters of the package were set to solvecomplex optimization problems (Xiang et al.2017). Therefore, these values were used tooptimize the pilot point values in this research.

METHODOLOGY

The methodology of the proposed approachincludes several steps to generate an optimum ore/waste boundary. The first step is to create anensemble of equiprobable realizations representingthe uncertainty on the position of the ore/wasteinterface. This step is followed by locating somepilot points in the simulation grid and fitting asmooth surface to them. The elevation values of thepilot points are then iterated and updated using theSA to seek the combination of the pilot point values

minimizing the total losses in each of the realiza-tions. These steps are illustrated in Figure 2. Moreinformation about the steps is given in the followingsubsections.

Simulations of the Bauxite/Ferricrete Contact

The proposed methodology requires an

ensemble of k conditional realizations R ¼Rj

��j ¼ 1; 2; . . . ; k� �

representing the ore/waste

interface generated as a first step. To perform thesimulations, the borehole elevations of the geologi-cal contact were used as the conditioning data.Available GPR survey of the area, on the otherhand, was used as the secondary information toguide the simulations. Creating such realizationsrather than a single estimation plays an importantrole in integrating the uncertainty in the designedexcavation surface.

The required simulations in this research wereobtained using the DS MPS algorithm due to thebenefits it provides in modeling the ore/wasteboundaries of lateritic metal deposits. For instance,it utilizes a TI as a structural model, rather than avariogram. Therefore, knowledge on the spatialstructures can be inferred from the previouslymined-out areas through the TI concept. Since themined-out topographies represent a complete pic-ture of the geological variations inherent in thecontact, they can provide rich structural information.In classical geostatistics, such information is derivedfrom sparse borehole data, which only offer partialknowledge of the ground truth. An additional ben-efit of using a TI is that the resulting modelingframework is rather non-expert friendly as vari-ogram modeling is not needed. Furthermore, the DSallows performing multivariate simulations by uti-lizing the multiple-point dependence between mul-tiple images. If geophysical data are available, as inour case, this can be incorporated easily in themodeling.

Although the use of MPS offers some benefits,the requirement of a TI to perform the simulationsmight sometimes limit its application. For instance,after the extraction of a bauxite deposit, a mined-outsurface is exposed and this can be used as a TIthrough a topographic survey. However, such surveydata may not always be readily available. In such


cases, the contact simulations can be performedusing standard geostatistical simulation techniquesas well. The DS MPS algorithm has been used in thisresearch since a mined-out floor survey (TI) wasalready available.

Locating Pilot Points

Once a set of realizations for the footwalltopography is generated, the next step involveslocating several pilot points h ¼ hðxlÞjf l ¼1; 2; . . . ;mg in the mining area. These pilot pointsfunction as synthetic elevation values, which areused to create an optimum ore/mining boundarythrough interpolation.

The process required to set up the pilot pointscan be explained in four steps, as illustrated in Fig-ure 3. The first one of these is to create a grid tostore their values and locations. The resolution ofthis grid can be chosen to be the same as the SG.Once this is done, the next step comprises locatingthe pilot points based on a predefined spacing. Inorder to better observe the effects of the chosenspacing on the results in this study, pilot points wereregularly spaced. That is, if the spacing is chosen asfive grid nodes, the pilot points are located at every5th node of the pilot points grid. After this step, theinitial values of the pilot points (synthetic elevation

values) need to be assigned. This can either beperformed by drawing numbers from a randomnumber generator or using the simulations. Therandom values for the pilot points can be generatedwithin a defined upper and lower boundaries. Suchboundaries can be determined using the maximumelevation value of the surface topography and theminimum elevation value of the contact realizations.Getting the initial values from the simulations cansimply be achieved by copying the elevation valuesfrom the nodes of a realization, which are co-locatedwith the pilot points. The final pilot point values, onthe other hand, are decided by the SA algorithmiteratively and lead to minimized dilution and orelosses. If the boundaries of the grid do not have atleast one pilot point, additional pilot points are alsoplaced at the boundaries. These additional pointsare required to make the interpolation cover thewhole modeling domain. For example, when locat-ing the pilot points in Figure 3, no points wereplaced in the right, left and bottom boundaries ini-tially. Therefore, three random locations in eachboundary were chosen to place an additional threepilot points.

It should be noted that the spacing chosen be-tween the pilot points affect the smoothness of thecreated mining boundaries as well as the dilution/oreloss amounts. If the spacing between the pilot pointsis small, the resulting surface becomes more de-

(a)Generate an ensemble of k realizations

using the DeeSseSurface

Topography

Realisations for thefootwall contact

Pilot Points

(b)Locate the pilot points in the simulation

domain

(c)Interpolate the pilot points to create a candidate mining

boundary and iterate the pilot point values

(d)Find the optimum values for the pilot points yielding the mining boundaries which minimize the sum of

expected losses in each realizationMultilevel B-Spline Surface

Synthetic elevation values

Figure 2. The main steps of the proposed methodology. See text for detailed explanations.


tailed. Therefore, it is advised to determine thisnumber based on the equipment flexibility as well. Asurface created using dense pilot points would yieldan uneven surface, which would increase the timeand fuel consumption required to perform theexcavation task.

Smooth Excavation Surface Design

Multilevel B-spline was used as an interpolationtechnique to construct the smooth excavation sur-face. The construction of the surface is mainly

accomplished by interpolating the Z field at eachgrid node of the mine area using a number h of pilotpoints. The degree of fluctuations that the resultingsurface exhibits is primarily influenced by two fac-tors. The first one is the number of h levels used inthe MBS interpolation. As this number increases,the fluctuations of the constructed surface also in-creases due to better approximations made in thefiner levels. The second factor is related to thespacing between the pilot points. Small spacingvalues between pilot points result in an increasednumber of pilot points. This would then lead togreater variations in the interpolated surface. We

Define the spacings between the pilot points in terms of grid spacing

Spacing inX direction

Spacing inY direction

Step 2Construct a grid for the pilot points.

Use the simulation grid dimensions

Step 1

Pilot Points Grid

Locate the pilot points based onthe chosen spacings between them

Pilot pointsPilot points at the boundaries

Step 3

Simulation Grid

Step 4

A realization for the

contact topography

Pilot points grid

Retrieve initial pilot point values from one of the realizations

Pilot points placed at every 4th grid node

Figure 3. Steps followed to locate the pilot points and assign their initial values in the mining area.


consider the fluctuations of the resulting surface asan essential factor for the equipment flexibility.Being able to adjust this allows one to integrate theequipment flexibility in the designed excavationsurface.

Loss Calculation-Objective Function

The objective of the optimization is to find theh ¼ hðxlÞjl ¼ 1; 2; . . . ;mf g pilot point values, which

lead to the decision surface SdðhÞ that minimizes thesum of expected economical losses in an ensemble ofk realizations:

minh

Xk

j¼1

LjðhÞ ð1Þ

where LjðhÞ is the loss incurred in the realization Rj

due to the decision surface SdðhÞ. It can be calcu-lated as follows:

LjðhÞ ¼ pmaxj � pactj ðhÞ ð2Þ

where pmaxj represents the maximum profit that can

potentially be made if all the ore between the sur-face topography and the ore/waste contact of the jthrealization were extracted. It can be calculated bymultiplying the unit profit P by the extracted vol-ume:

pmaxj ðhÞ ¼ P

Xn

i¼1

Tmaxi;j ðhÞ �A ð3Þ

where Tmaxi;j represents the maximum bauxite thick-

ness at the ith grid node of the jth realization (seeFig. 4), A represents the area of a grid cell and n rep-resents the number of informed grid nodes in thesimulation area. Tmax

i;j can simply be calculated by

subtracting the elevation of the footwall topography

realisations from thoseof the surface topographyZtopoi :

Tmaxi;j ¼ Z

topoi � Ri;j ð4Þ

pactj ðhÞ, on the other hand, represents the actual

profit that can be made out of Rj if the mining is

carried out following the boundaries defined by the

decision surface SdðhÞ. Its calculation is performedby subtracting the cost of dilution from the profitmade out of extracting the ore at each grid node:

pactj ðhÞ ¼ PXn

i¼1

Tbaui;j ðhÞ �A� C

Xn

i¼1

Tdili;j ðhÞ �A ð5Þ

where Tbaui;j represents the mined bauxite (ore)

thickness using the decision surface SdðhÞ, Tdili;j rep-

resents the ferricrete (waste) thickness overlying thedecision surface and C represents the unit cost ofdilution. These thicknesses can be calculated asfollows:

Tbaui;j ðhÞ ¼Z

topoi �maxðRi;j;minðZtopo

i ; Sdi ðhÞÞÞ ð6Þ

Tdili;j ðhÞ ¼Ri;j �minðRi;j;minðZtopo

i ; Sdi ðhÞÞÞ ð7Þ

To sum up, the objective function used for theoptimization was evaluated based on the expected

Topography

Simulated ore/waste contact

Excava�on surface fi�ed to the pilot points

Bauxite unit

Ferricrete unit

bauT

dilT

maxT maxTbauT

Pilot points

Figure 4. Thicknesses used to calculate the losses due to an excavation surface.


losses incurred due to a decision surface SdðhÞ. Itscalculation was performed in four steps: (1) gener-ation of an ensemble of realizations (only for once),(2) generating a set of pilot points, (3) fitting anMBS surface to the pilot points and (4) evaluatingthe loss due to the constructed smooth surface ineach of the stochastic realizations.

Determination of Optimum Values at the PilotPoints

The pilot point values yielding an excavationsurface that minimizes the expected loss weredetermined by the optimization framework of theSA algorithm. The steps of the optimization frame-work to design the optimal mining boundaries canbe seen in Figure 5.

Given a set of initial h pilot points (vector ofdecision variables), an MBS surface is first fitted tothem, and the sum of losses in all the realizations iscalculated. The SA algorithm then perturbs the pilotpoint values using the Cauchy–Lorentz visiting dis-tribution to evaluate the performance of a newsolution (losses caused by the updated pilot pointvalues). A change in the pilot point values yieldingan improvement in the objective function (reductionin the losses) is always accepted. On the other hand,

any change in the pilot point values resulting in aworse solution (increase in the loss) can be acceptedor rejected based on the probability calculated usingthe generalized Metropolis algorithm. The accep-tance probability depends on the artificial tempera-ture parameter of the SA. As the temperature set ishigh in the initial stages of the optimization process,the probability of accepting worse solutions is highas well. Therefore, the solution space is well ex-plored in the beginning. As the iterations progress,the probability of accepting a worse solution goesdown since the artificial temperature approaches tozero. After several thousands of iterations, the SAconverges and finds the hopt pilot point values min-

imizing the losses. Once the optimum pilot pointvalues are found, they are then used to design anoptimum excavation surface through the MBSmethod.

RESULTS AND DISCUSSION

The proposed approach was implemented togenerate optimum mining boundaries for a laterite-type bauxite deposit. Being an initial step of theproposed methodology, simulations of the baux-ite/ferricrete interface were first performed. In orderto achieve this, the elevation variable of the inter-

Interpolate the pilot points using theMu -level B-Spline

Candidate ore boundaries

Realiza onsMPS Simula

Evaluate theobjec ve func(expected losses)

In values for the pilot points

Start

Calculate the energy difference

Is the energy difference

lower?

Op mumPilot Point

Values

ΔE

[ ]E L

Is the probability of acceptance

> random

number

Yes

Accept new iteration

No

Yes

Propose new pilot point

values

Is the stopping

criterion met?No

No

Yes

Create the optimum mining

boundaries

Figure 5. Steps used to determine the pilot points yielding optimum mining boundaries.


face was used as the attribute to be simulated. Dueto the existence of both borehole and GPR data, thesimulations were performed in the form of bivariatesimulations. This was carried out by utilizing theborehole data as the primary variable to conditionthe simulations and GPR data, which are exhaus-tively sampled throughout the simulation domain, asthe secondary information to guide the simulations.

Both the borehole and GPR data contain theelevation variable of the bauxite/ferricrete interface.The borehole elevations were obtained by observingthe elevation values at which the lithology changesfrom bauxite to ferricrete. The GPR elevations, onthe other hand, were obtained indirectly from theoriginal raw GPR measurements. In the first place,the raw GPR data were acquired in two-way traveltime. Therefore, it initially allowed the determina-tion of the depth from the surface to the baux-ite/ferricrete interface. After subtracting the depthsto the interface from the surface elevations, theGPR elevations for the bauxite/ferricrete were ob-tained. These elevations were used as the secondaryvariable to guide the simulations. The conditioningdata used in the simulations are shown in Figure 6.

A bivariate TI was then constructed to infer themultiple-point dependence between the boreholeand the GPR data. Its variables comprise an exposedmined-out surface of a previously extracted miningarea and an extensive GPR survey conducted beforemining. The variables of the constructed TI can beseen in Figure 7.

The grid used to store the TI dataset consists of180 nodes in easting (X) and 400 nodes in northing(Y) directions. On the other hand, the SG is com-prised of 97 and 214 nodes in both easting andnorthing directions, respectively. The single grid sizeis defined as 2.38 m 9 2.38 m for both the TI andthe SG grids. Since the original data for the GPRand the mined-out floor surface were in the form ofpoint data, they were migrated to the TI grid nodelocations. This was performed through the condi-tional sequential Gaussian simulation (sGs) tech-nique so as to preserve the original statisticalproperties as well as avoiding any smoothing effect.Using the constructed TI, the DS was used to gen-erate an ensemble of 200 realizations. The averageof the resulting simulations can be seen in Figure 8.

The pilot points were placed in the simulationdomain based on a defined grid spacing betweenthem. In order to analyze the effect of spacing on thelosses and the fluctuations of the decision surface,

pilot points spaced 8, 16 and 24 number of grids weretested. The defined spacings yielded 251, 60 and 26pilot points, respectively. Plan views of the pilotpoint locations in the mining area can be seen inFigure 9.

In addition to the spacings between the pilotpoints, the number of h levels used in the MBSmethod also affects the smoothness of the decisionsurface. This parameter was chosen as 10 in thisstudy based on visually inspecting the smoothness ofthe resulting surface. Although our choice in thisresearch was mainly due to the visual inspection, wesuggest that a suitable value of this parameter bedetermined in the future to yield a design surfacemimicking the front-end loader equipment selectiv-ity.

The loss calculations require some unit costsand profits be defined. These include the profit P ofmining a unit volume of bauxite ore and the cost Cincurring in the case of a unit volume of dilution.Since the grade distribution does not show a signif-icant variability throughout the deposit, we simply

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and the underlying image represents the GPR data (secondary

variable).


assume that the Al2O3% grades overlying the ore/waste interface are the same everywhere. A similarassumption was also made for the SiO2% grades

within the ferricrete unit. Therefore, given thatdilution is approximately 60 times costlier than oreloss in the mining of such deposits (Erten 2012), wesimply assumed that a unit loss occurring due todilution basically costs $60. We also assumed thatthe profit made when one unit of ore is mined is $1.It should be noted that these prices are hypotheticaland might not reflect the reality.

The optimization process begins with assigningan initial set of values for the pilot points to beoptimized. These values can either be randomlychosen or pre-specified before the optimization. Inour case, the values were taken from the elevationvalues corresponding to the average of 200 realiza-tions at the pilot point locations. We also defined alower and an upper boundary in which the optimumvalues of the pilot points are sought. These bound-aries function as the constraints of the optimization.The lower boundary was calculated based on theminimum elevation value of the bauxite/ferricreterealizations. Maximum elevation constraint, on theother hand, was the maximum elevation value of thetopography. We defined the maximum iterationnumber of the SA as 50,000 and the temperature as5000. An objective function call during the opti-

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Figure 7. The constructed bivariate TI: (a) extraction surface of a previously mined-out area and (b) extensive

GPR survey carried out prior to the excavation of the mining area.

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bauxite/ferricrete interface.


mization process leads to the loss calculation for 200images, which were of size 97� 214, to calculate thelosses in each of the realizations. Having ran the SAusing the defined setup, it converged and yielded theoptimum pilot point values. The resulting crosssections of the deposit for a different number ofpilot points are shown in Figures 10 and 11. Planviews of the generated smooth surfaces are shown inFigure 12.

The cross sections demonstrate that the opti-mum surfaces constructed lie above most of therealizations. This is mainly due to the introduction ofa higher dilution cost compared to that of ore loss.The proposed approach automatically avoids gen-erating a surface that causes dilution, as it leads togreater losses in the objective function. Note that theposition of the decision surface may seem to be highabove the simulations in certain cases, but this canbe since we are only looking at a section while thereare fluctuations in the perpendicular direction thatthe decision surface needs to consider to remainoptimal.

The use of a different number of pilot pointshas two main consequences. The first one is about

the fluctuations seen in the decision surface gener-ated. When the number of pilot points used in-creases, the decision surface exhibits morefluctuations. Similarly, the use of sparser pilot pointsyields decision surfaces that exhibit less fluctuations,as can be seen in Figure 12. The second consequenceis that the calculated losses decrease when thenumber of pilot points increases as shown in Fig-ure 13. This indicates that there is a trade-off be-tween the fluctuations and the resulting losses. Morepilot points allow defining a design surface that willbe rougher and more difficult to excavate but willproduce higher revenue.

Following the collection of the borehole andGPR data, the area was mined out by the front-endloader operator utilizing the hardness differencebetween the ore/waste to track the actual geologicalinterface. The surface exposed was mapped througha topographic survey, and the collected surveypoints were then used to create the complete imageof the mined-out surface. Point to grid data con-version has been achieved by the conditional sGs, asin the construction of the TI. The main idea wasagain to prevent any smoothing effect.

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Figure 9. Locations of the pilot points in the mine area. Blue dots represent the pilot points located at every (a)

8th grid node, (b) 16th grid node and (c) 24th grid node. The dashed lines represent the coordinates where the

cross sections were taken as presented in Figures 10 and 11.


In order to make a comparison, the expectedvolumes for the bauxite reserve, mined portion ofthe reserve, dilution and ore losses were calculatedusing the three optimized boundaries. The ex-pected reserve volume was calculated by takingthe average of the volume between the surfacetopography and 200 contact realizations. For themined reserve calculations, the bauxite volumeoverlying the optimized surfaces in 200 realizationswere averaged. If the proposed surfaces were be-low a realization at a grid node, the elevationdifferences were multiplied by the area to calcu-late the dilution volume. The sum of all the dilu-tion at each grid node yielded the total dilutionamount for a realization. Similarly, if the opti-

mized surfaces were above a realization, they wereconsidered to cause an ore loss and the associatedvolumes were calculated for each of the realiza-tions. The dilution and ore losses calculated for200 realizations were then averaged to find theexpected dilution and ore losses for a given deci-sion surface. In addition, the results of theseoptimized surfaces were also compared with themined-out surface using the same calculation logic.The summary of the volume calculations given inTable 1 demonstrates the benefit of the proposedmethod. Concerning the volume calculations, thebauxite mined using 251 pilot points comprises77% of the expected bauxite reserve. This is verysimilar to the amount of bauxite mined by the

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Figure 10. Y–Y cross sections of the optimum surfaces found by using (a) 251 pilot points, (b) 60 pilot points and

(c) 26 pilot points.


operator, which was 76% of the deposit. However,although both of the surfaces result in obtainingthe similar amount of bauxite deposit, the dilutionamount resulted using 251 pilot point surface was0.076% of the total mined volume and significantlylower than the dilution amount of mined-out sur-face, which was 2.7%. The percentages describedare also illustrated in Figure 14 in terms of bar-plots.

SUMMARY AND CONCLUSIONS

In this paper, we presented a new grade controltechnique tominimize the risk of operational dilution

andore losses in lateriticmetal deposits.Although thedevelopment was performed on a lateritic metal de-posit, the method can be applied to any stratified de-posit to create an ore/waste boundary with a certaindegree of smoothness. The proposed approach ben-efits from a parameter calibration technique called‘‘pilot points’’ to create a design surface with multi-level B-spline method. The optimized pilot pointvalues are iteratively obtained within the simulatedannealing algorithm to create a new ore/wasteboundary minimizing the risk of dilution and orelosses. Possible losses of a constructed surface arecalculated using several scenarios of the ore/wasteboundaries generated by themultiple-point statistical

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Figure 11. X–X cross sections of the optimum surfaces found by using (a) 251 pilot points, (b) 60 pilot points and

(c) 26 pilot points.


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Figure 12. Plan view of the excavation surface designed using (a) 251 pilot points, (b) 60 pilot points, (c) 26 pilot

points and (d) the actual mined-out surface. The dashed lines show the sections where the cross sections in

Figures 10 and 11 were constructed.


simulations. We have implemented the proposedapproach on a lateritic bauxite deposit and comparedthe resulting losses with the ones calculated using theactual mining operation.

The major advantage of the proposed method isthe reduction in the economical losses. Implemen-tation of the method on bauxite has demonstrated

much less losses compared to the actual miningoperation that took place. It was also observed thatthe losses resulting from our proposed approach areaffected by the spacing between the pilot points.Densely spaced pilot points give smaller losses butincrease the fluctuations in the resulting surface.Another benefit of the proposed approach com-

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Figure 13. Histograms of the losses calculated using the (a) 251 pilot points, (b) 60 pilot points, c 26 pilot points

and (d) the actual mined-out surface.

Table 1. Statistics of the proposed and mined-out surfaces

Expected stats 251 Pilot points 60 Pilot points 26 Pilot points Mined out

Reserve volume ðm3Þ 130,448 130,448 130,448 130,448

Mined reserve ðm3Þ 100,506 86,938 75,324 99,316

Ore loss ðm3Þ 29,942 43,510 55,124 31,131

Ferricrete dilution ðm3Þ 76 86 113 2791

Economical losses $34,502 $48,713 $61,946 $159,104


prises the integration of the uncertainties in the ore/waste contact. The losses of a decision surface arecalculated using the equiprobable realizations rep-resenting the ore/waste interface. Therefore, theuncertainty in the ore/waste boundary is accountedfor by the design surface. Although the realizationswere generated using multiple-point statistics, themethodology could also work well with standardgeostatistical simulations, such as Turning Bands.One should, however, be cautious of the quality ofthe simulations used as it significantly affects thedesigned surface. High variability in the ore/wastecontact simulations, for instance, tends to result in adesign surface deviating away from the average ofthe realizations due to the high penalty associatedwith the dilution. This can lead to the underesti-mation of the mineable reserve volume.

The last benefit is about the adjustable smooth-ness of the generated surface. The number of h le-vels of the multilevel B-spline method allowsconstructing surfaces with a varying degree ofsmoothness. This can help designing surfaces whichare capable of reflecting the mining equipmentselectivity. Although h parameter of the multilevelB-spline fundamentally controls the smoothness ofthe surface, it needs to be calibrated in conjunctionwith the pilot point spacing used, as it also plays acrucial role on the fluctuations seen in the resultingsurface.

The proposed approach reveals several pointsto be studied in the future as an improvement. The

first one is the subjectivity introduced when placingthe pilot points. The number and the locations of thepilot points are chosen based on personal preferencein this study. Therefore, automatic determination ofthe number and the location of the pilot points as inJimenez et al. (2016) may eliminate the subjectivityintroduced. The second point that can be improvedis related to the degree of smoothness that thedecision surface exhibits. Since there is a trade-offbetween the losses and the degree of smoothness ofthe decision surface, the optimum degree ofsmoothness yielding minimum losses needs to bespecified. This can be achieved by establishing arelationship between the surface smoothness and themining equipment-related losses. Once such rela-tionship is formulated as a function of the fluctua-tions of a given surface, this can then be used in theobjective function as an additional term to calculatethe total losses. Lastly, although the computationtime required to perform the optimization was rea-sonable (6–8 h) using the R software, it can signifi-cantly be reduced by utilizing parallel computingand coding in C language.

Instead of using the multilevel B-Spline, alter-native interpolation techniques, such as kriging, canbe used to generate the excavation surface. Thebenefit of kriging would be the possibility to adjustthe smoothness of the surface with the rangeparameter of the variogram model. In addition, itcould be possible to infer this range from previouslyexcavated surfaces. Therefore, the equipment flexi-

251 PP 60 PP 26 PP Mined Out

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Figure 14. Barplots representing (a) the mined bauxite ore percentage of the expected reserve and (b) dilutionpercentages of the total mined volumes.


bility can automatically be integrated into the de-signed surface with the prior knowledge from themined-out areas. The implementation of multilevelB-splines in this research was due to its high-speedcomputation (Saveliev et al. 2005). Future researchcould investigate the use of kriging for the interpo-lation and explore its possible advantages.

Due to the fairly continuous nature of Al2O3%and SiO2% grades throughout the deposit and alsofor the sake of simplicity, the grades were consid-ered constant in this study. Therefore, in order tosqueeze more performance out of the approach andalso to better reflect the reality, a block model ofthese attributes can also be constructed to calculatethe losses. Use of such a model would then involvethe loss calculations based on the partial or completemining of a specific block.

ACKNOWLEDGMENTS

The authors would like to thank the editor andthree anonymous reviewers for their valuable com-ments and suggestions.

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