# Indicator Variograms as an Aid for Geological Interpretation and Modeling of Ore Deposits

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Indicator Modeling

_{k}, with k = 1, …, K, can be codified by a binary variable called indicator:

**x**stands for a generic point in the three-dimensional Euclidean space R

^{3}. Viewing this indicator as a spatial random field in R

^{3}opens the way for the use of the geostatistical formalism, which allows for prediction or simulation of the indicator at any point based on the values observed at surrounding sampling points.

_{k}(

**x**) represents the probability of finding domain D

_{k}at point

**x**:

_{k}does not depend on

**x**and stands for the proportion of space covered by domain D

_{k}. First-order stationarity means that the probability of finding domain D

_{k}is constant over space.

#### 2.2. Theoretical Indicator Direct Variogram

_{k}is the direct variogram of the indicator random field I

_{k}, defined for each pair of points (

**x**,

**x**′) as:

_{k}is second-order stationary, ${\gamma}_{k}$ only depends on the separation vector between

**x**and

**x**′ and is commonly denoted as ${\gamma}_{k}(h)$ with

**h**=

**x**−

**x**′. The indicator direct variogram ${\gamma}_{k}(h)$ represents the probability of transitioning from inside D

_{k}at point

**x**to outside D

_{k}at point

**x**′ =

**x**−

**h**(equivalently, from outside D

_{k}at point

**x**to inside D

_{k}at point

**x**−

**h**), the transition happening somewhere over the extent of the separation vector

**h**.

**x**−

**x**′ becomes infinitely large, the indicator variables I

_{k}(

**x**) and I

_{k}(

**x**′) are independent, so that $P\{x\text{}\in \text{}{D}_{k},{x}^{\prime}\text{}\notin \text{}{D}_{k}\}=P\{x\text{}\in \text{}{D}_{k}\}\text{\hspace{0.17em}}P\{{x}^{\prime}\text{}\notin \text{}{D}_{k}\}={p}_{k}(1-{p}_{k})$. For separation vectors with a large norm, the indicator direct variogram presents a sill, whose value relates to the proportion of space (p

_{k}) covered by domain D

_{k}:

_{k}(

**x**) and I

_{k}(

**x**+

**h**) are independent. This distance may depend on the direction of

**h**and does not relate to the size of domain D

_{k}[9] (under the first-order stationarity assumption, D

_{k}has a constant probability of occurrence over R

^{3}, so it is unbounded) nor to the size of its convex parts [16] (based on Equation (5), D

_{k}has the same indicator direct variogram as its complement, whose convex parts differ from that of D

_{k}).

_{k}can be obtained from the indicator direct variogram, in particular, from its behavior at the origin. The first-order derivative conveys information about the regularity of the boundary of D

_{k}[13,17]:

_{k}per unit volume of R

^{3}, S

_{3}is the unit sphere of R

^{3},

**u**a unit-length vector and ${{\gamma}_{k}^{\prime}}^{(u)}(0)$ is the first-order derivative of the variogram at the origin in the direction of vector

**u**:

_{k}is void (p

_{k}= 0) or fills the whole space R

^{3}(p

_{k}= 1), a null derivative (indicator direct variogram with a zero slope at the origin) in every direction cannot be observed [18]. A finite non-zero derivative means that the boundary of D

_{k}is almost always regular (Figure 1A,B), while an infinite derivative means that D

_{k}has an irregular or fractal boundary (Figure 1C,D). In the former case, the second-order derivative of the indicator direct variogram at the origin in the direction of vector

**u**relates to the existence of singular points along the boundary of D

_{k}[19]:

_{k}has a smooth boundary everywhere), then the second-order derivative of the indicator direct variogram at the origin is zero, which means that the variogram is straight at the origin (Figure 2C,D).

_{k}. In such a case, the geometric properties of domain D

_{k}and of its boundary are not the same in all the directions of space.

#### 2.3. Experimental Indicator Direct Variogram

^{+}(

**h**) = {$(\alpha ,\beta )\in {\{1,\dots n\}}^{2}:{x}_{\alpha}-{x}_{\beta}=h$ up to distance and angle tolerances} and card{N

^{+}(

**h**)} is the cardinality of N

^{+}(

**h**). Under a second-order stationarity assumption, this is an unbiased estimator of the indicator direct variogram ${\gamma}_{k}(h)$, so that its properties (behavior at large distances, behavior near the origin, anisotropy) should be the same, up to statistical fluctuations.

_{k}is not constant in space (presence of a systematic trend or zonation, which can be explained by geological considerations). In such a situation, one may assume that the second-order stationarity assumption is valid only at small scales (local stationarity), and restrict the interpretation of the experimental variogram to small distances. In particular, the regularity and smoothness of the domain boundary can still be inferred from the behavior of the variogram near the origin.

#### 2.4. Theoretical Indicator Cross-Variogram

_{k}and D

_{k}

_{′}that do not overlap. The joint behavior of these two domains can be modeled by the cross-variogram of the associated indicator fields I

_{k}and I

_{k}

_{′}:

_{k}to D

_{k}

_{′}(or vice-versa, from D

_{k}

_{′}to D

_{k}) between two points separated by vector

**h**. If the two indicator random fields I

_{k}and I

_{k}

_{′}are jointly second-order stationary, ${\gamma}_{k,{k}^{\prime}}$ only depends on

**h**=

**x**–

**x**′ and is commonly denoted as ${\gamma}_{k,{k}^{\prime}}(h)$.

**x**−

**x**′ becomes infinitely large, the indicator variables I

_{k}(

**x**) and I

_{k}

_{′}(

**x**) are independent of I

_{k}(

**x**′) and I

_{k}

_{′}(

**x**′), so that [14]:

_{k}and D

_{k}

_{′}. The distance $\Vert h\Vert $ at which the sill is reached (correlation range) corresponds to the distance beyond which the indicator variables I

_{k}(

**x**) and I

_{k}

_{′}(

**x**) are independent of I

_{k}(

**x**+

**h**) and I

_{k}

_{′}(

**x**+

**h**). This distance may differ from the correlation ranges of the indicator direct variograms and may also depend on the direction of

**h**. Note that the two domains D

_{k}and D

_{k}

_{′}cannot be independent, insofar as they are assumed not to overlap; as a consequence, the cross-variogram cannot be identically zero, which is corroborated by the existence of a negative sill (Equation (13)), unless one of the domains is void (p

_{k}= 0 or p

_{k}

_{′}= 0).

_{k}and D

_{k}

_{′}. In particular, one has:

_{k}and D

_{k}

_{′}per unit volume of R

^{3}and ${\gamma}_{k,{k}^{\prime}}^{{}^{\prime}(u)}(0)$ is the first-order derivative of the indicator cross-variogram at the origin in the direction of vector

**u**:

_{k}and D

_{k}

_{′}have no contact (Figure 3A,B). Even more, the distance up to which the cross-variogram is equal to zero in a given direction represents the minimum separation distance between the two domains D

_{k}and D

_{k}

_{′}along this direction. In contrast, an indicator cross-variogram with a negative derivative (negative slope) at the origin indicates the existence of contacts between D

_{k}and D

_{k}

_{′}: the greater the slope, the more contact area exists (Figure 3C–F).

#### 2.5. Experimental Indicator Cross-Variogram

_{k}and D

_{k}

_{′}are bounded, the experimental indicator cross-variogram will be zero for distances larger than the maximum separation distance between points from the two domains (all the terms in the summation of Equation (16) are equal to zero). As stated in Section 2.3, one may then assume that the second-order stationarity assumption is valid only locally and restrict the interpretation of the experimental indicator cross-variogram to small distances. In particular, the slope at the origin reflects the amount of contact area between D

_{k}and D

_{k}

_{′}.

#### 2.6. Link between Indicator Direct and Cross-Variograms

#### 2.7. Ratio of Indicator Cross-to-Direct Variograms and Edge Effects

_{k}

_{′}at point

**x**′ when leaving D

_{k}at point

**x**or, vice-versa, of reaching domain D

_{k}

_{′}at point

**x**when leaving D

_{k}at point

**x**′:

**x**’ belongs to D

_{k}

_{′}knowing that it does not belong to D

_{k}, which is equal to the proportion of domain D

_{k}

_{′}relative to the proportion of what is not D

_{k}. Such a sill value can be used as a reference to determine whether or not the transition from domain D

_{k}to domain D

_{k}

_{′}between two points separated by vector

**h**or −

**h**is preferential, a situation known as an edge effect or a border effect [14,20]. In particular, for small separation vectors ($h=\mathsf{\epsilon}u$ with

**u**a unit vector and $\mathsf{\epsilon}$ a scalar tending to zero), the following cases may be found (Figure 4):

- $-\frac{{\gamma}_{k,{k}^{\prime}}(\mathsf{\epsilon}u)}{{\gamma}_{k}(\mathsf{\epsilon}u)}<\frac{{p}_{{k}^{\prime}}}{1-{p}_{k}}$: due to Equations (19) and (20), $P\{{x}^{\prime}\text{}\in \text{}{D}_{{k}^{\prime}}|x\text{}\in \text{}{D}_{k},{x}^{\prime}\text{}\notin \text{}{D}_{k}\}P\{{x}^{\prime}\text{}\in \text{}{D}_{{k}^{\prime}}|{x}^{\prime}\text{}\notin \text{}{D}_{k}\}$ where ${x}^{\prime}=x+\mathsf{\delta}\mathsf{\epsilon}u$ is a point located close to
**x**. Knowing that**x**belongs to D_{k}decreases the probability that**x**’ belongs to D_{k}_{′}, given that it does not belong to D_{k}This means that the contact area between D_{k}and D_{k}_{′}is smaller than it would be in the absence of edge effect, i.e., there is a propensity for both domains not to be in contact (Figure 4A,B). - $-\frac{{\gamma}_{k,{k}^{\prime}}(\mathsf{\epsilon}u)}{{\gamma}_{k}(\mathsf{\epsilon}u)}=\frac{{p}_{{k}^{\prime}}}{1-{p}_{k}}$: the contact area between domains D
_{k}and D_{k}_{′}coincides with the expected contact area in the absence of edge effect (no preferential contact) (Figure 4C,D). - $-\frac{{\gamma}_{k,{k}^{\prime}}(\mathsf{\epsilon}u)}{{\gamma}_{k}(\mathsf{\epsilon}u)}>\frac{{p}_{{k}^{\prime}}}{1-{p}_{k}}$: the contact area between D
_{k}and D_{k}_{′}is larger than it would be in the absence of edge effect. There is a propensity for both domains to be in contact together (Figure 4E,F).

_{k}and D

_{k}

_{′}necessarily implies the existence of an opposite edge effect between D

_{k}and D

_{k}

_{″}for some k″ $\in $ {1, …, n}.

## 3. Case Study and Results

#### 3.1. Presentation of the Deposit

- Andesite (code 1): this volcanic rock is distributed as an interbedded sequence spread over the whole deposit and consists of lavas, auto-breccias and ocoites, with porphyritic to aphanitic texture and, locally, with amygdaloidal to ocoitic texture.
- Other volcanic rocks (code 2): this is a sequence of differentiated volcanic rocks with textures that vary from porphyritic to tuffaceous with an aphanitic matrix.
- Sedimentary rocks (code 3): this rock type includes undifferentiated, volcaniclastic and calcareous sediments, which are mostly interbedded with dacite and andesite, forming the host rock of the intrusive porphyry.
- Granodioritic porphyry (code 4): this is a medium grain granodiorite of Triassic age.
- Monzonitic porphyry (code 5): this porphyry rock is associated with the hypogene mineralization.
- Hydrothermal breccia (code 6) with tourmaline, quartz, chalcopyrite and pyrite cements, which can be found in the contact between the monzonitic porphyry with the host rock as a breccia with intrusive fragments.

#### 3.2. Data Presentation and Pre-Processing

#### 3.3. Indicator Direct Variograms

#### 3.3.1. Existence of a Sill

#### 3.3.2. Shape and Anisotropy

#### 3.3.3. Slope at the Origin

#### 3.4. Indicator Cross-Variograms

#### 3.4.1. Behavior at Large Distances

#### 3.4.2. Behavior at Short Distances

#### 3.5. Ratios of Indicator Cross-to-Direct Variograms

## 4. Discussion

#### 4.1. Geological Interpretation and Modeling

#### 4.2. Geostatistical Modeling and Simulation

## 5. Conclusions

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## Appendix A

_{k}and D

_{k}

_{′}that do not overlap. The non-centered cross-covariance of the domains indicators I

_{k}and I

_{k}

_{′}is defined as:

_{k}

_{,k′}only depends on

**h**=

**x**′ −

**x**and tends to p

_{k}p

_{k}

_{′}when the norm of

**x**′ −

**x**becomes infinitely large. Furthermore, the boundary common to both domains D

_{k}and D

_{k}

_{′}is related with the average derivative of C

_{k}

_{,k′}at the origin (Equation (14)):

_{k}and D

_{k}

_{′}and on asymmetrical relationships between these domains (for instance, D

_{k}being preferentially above D

_{k}

_{′}rather than below), a pattern that can sometimes be explained by chemical or sedimentological processes [9,31,32]. In particular, the ratio between indicator cross-covariance and direct variogram is:

_{k}

_{′}/(1 – p

_{k}) and represents the reference value against which one can determine the existence of directional edge effects at very small distances:

- $\frac{{C}_{k,{k}^{\prime}}(\mathsf{\epsilon}u)}{{\gamma}_{k}(\mathsf{\epsilon}u)}<\frac{{p}_{{k}^{\prime}}}{1-{p}_{k}}$: the contact area between D
_{k}and D_{k}_{′}when moving along vector**u**(i.e., leaving D_{k}at**x**and reaching D_{k}_{′}at $x+\mathsf{\epsilon}u$) is smaller than it would be in the absence of a directional edge effect. - $\frac{{C}_{k,{k}^{\prime}}(\mathsf{\epsilon}u)}{{\gamma}_{k}(\mathsf{\epsilon}u)}=\frac{{p}_{{k}^{\prime}}}{1-{p}_{k}}$: the contact area between D
_{k}and D_{k}_{′}when moving along vector**u**is equal to the expected area in the absence of a directional edge effect. - $\frac{{C}_{k,{k}^{\prime}}(\mathsf{\epsilon}u)}{{\gamma}_{k}(\mathsf{\epsilon}u)}>\frac{{p}_{{k}^{\prime}}}{1-{p}_{k}}$: the contact area between D
_{k}and D_{k}_{′}when moving along vector**u**is larger than it would be in the absence of an edge effect.

## References

- Dowd, P.A. Structural controls in the geostatistical simulation of mineral deposits. In Geostatistics Wollongong’96; Baafi, E.Y., Schofield, N.A., Eds.; Kluwer Academic: Dordrecht, The Netherlands, 1997; pp. 647–657. [Google Scholar]
- Vargas-Guzmán, J.A. Transitive geostatistics for stepwise modeling across boundaries between rock regions. Math. Geosci.
**2008**, 40, 861–873. [Google Scholar] [CrossRef] - Rossi, M.E.; Deutsch, C.V. Mineral Resource Estimation; Springer: London, UK, 2014; p. 332. [Google Scholar]
- Madani, N.; Emery, X. Plurigaussian modeling of geological domains based on the truncation of non-stationary Gaussian random fields. Stoch. Environ. Res. Risk Assess.
**2017**, 31, 893–913. [Google Scholar] [CrossRef] - Mery, N.; Emery, X.; Cáceres, A.; Ribeiro, D.; Cunha, E. Geostatistical modeling of the geological uncertainty in an iron ore deposit. Ore Geol. Rev.
**2017**, 88, 336–351. [Google Scholar] [CrossRef] - Matheron, G.; Beucher, H.; de Fouquet, C.; Galli, A.; Guérillot, D.; Ravenne, C. Conditional simulation of the geometry of fluvio-deltaic reservoirs. In Proceedings of the SPE Annual Technical Conference and Exhibition, Dallas, TX, USA, 27–30 September 1987; Society of Petroleum Engineers: Richardson, TX, USA, 1987; pp. 591–599. [Google Scholar]
- Ravenne, C.; Galli, A.; Doligez, B.; Beucher, H.; Eschard, R. Quantification of facies relationships via proportion curves. In Geostatistics Rio 2000; Armstrong, M., Bettini, C., Champigny, N., Galli, A., Remacre, A., Eds.; Kluwer: Dordrecht, The Netherlands, 2002; pp. 19–40. [Google Scholar]
- Xu, C.; Dowd, P.A.; Mardia, K.V.; Fowell, R.J. A flexible true plurigaussian code for spatial facies simulations. Comput. Geosci.
**2006**, 32, 1629–1645. [Google Scholar] [CrossRef] - Carle, S.F.; Fogg, G.E. Transition probability-based indicator geostatistics. Math. Geol.
**1996**, 28, 453–476. [Google Scholar] [CrossRef] - Carle, S.F.; Fogg, G.E. Modeling spatial variability with one- and multidimensional continuous-lag Markov chains. Math. Geol.
**1997**, 29, 891–918. [Google Scholar] [CrossRef] - Boisvert, J.B.; Pyrcz, M.J.; Deutsch, C.V. Multiple point metrics to assess categorical variable models. Nat. Resour. Res.
**2010**, 19, 165–175. [Google Scholar] [CrossRef] - Dimitrakopoulos, R.; Mustapha, H.; Gloaguen, E. High-order statistics of spatial random fields: Exploring spatial cumulants for modeling complex non-Gaussian and non-linear phenomena. Math. Geosci.
**2010**, 42, 65–99. [Google Scholar] [CrossRef] - Lantuéjoul, C. Geostatistical Simulation: Models and Algorithms; Springer: Berlin/Heidelberg, Germany, 2002; p. 256. [Google Scholar]
- Séguret, S.A. Analysis and estimation of multi-unit deposits: Application to a porphyry copper deposit. Math. Geosci.
**2013**, 45, 927–947. [Google Scholar] [CrossRef][Green Version] - Beucher, H.; Renard, D. Truncated Gaussian and derived methods. C. R. Geosci.
**2016**, 348, 510–519. [Google Scholar] [CrossRef] - Armstrong, M.; Galli, A.; Beucher, H.; Le Loc’h, G.; Renard, D.; Doligez, B.; Eschard, R.; Geffroy, F. Plurigaussian Simulations in Geosciences; Springer: Berlin/Heidelberg, Germany, 2011; p. 176. [Google Scholar]
- Matheron, G. Eléments Pour une Théorie des Milieux Poreux; Masson: Paris, France, 1967; p. 166. [Google Scholar]
- Dubrule, O. Indicator variogram models: Do we have much choice? Math. Geosci.
**2017**, 49, 441–465. [Google Scholar] [CrossRef] - Emery, X.; Lantuéjoul, C. Geometric covariograms, indicator variograms and boundaries of planar closed sets. Math. Geosci.
**2011**, 43, 905–927. [Google Scholar] [CrossRef] - Roth, C. Incorporating information about edge effects when simulating lithofacies. Math. Geol.
**2000**, 32, 277–300. [Google Scholar] [CrossRef] - Rivoirard, J. Weighted variograms. In Geostatistics 2000 Cape Town; Kleingeld, W.J., Krige, D.G., Eds.; Geostatistical Association of Southern Africa: Cape Town, South Africa, 2001; pp. 145–155. [Google Scholar]
- Emery, X.; Ortiz, J.M. Weighted sample variograms as a tool to better assess the spatial variability of soil properties. Geoderma
**2007**, 140, 81–89. [Google Scholar] [CrossRef] - De Souza, L.E.; Costa, J.F.C.L. Sample weighted variograms on the sequential indicator simulation of coal deposits. Int. J. Coal Geol.
**2013**, 112, 154–163. [Google Scholar] [CrossRef] - Emery, X. Reducing fluctuations in the sample variogram. Stoch. Environ. Res. Risk Assess.
**2007**, 21, 391–403. [Google Scholar] [CrossRef] - Clark, R.G.; Allingham, S. Robust resampling confidence intervals for empirical variograms. Math. Geosci.
**2011**, 43, 243–259. [Google Scholar] [CrossRef] - Olea, R.A.; Pardo-Igúzquiza, E. Generalized bootstrap method for assessment of uncertainty in semivariogram inference. Math. Geosci.
**2011**, 43, 203–228. [Google Scholar] [CrossRef] - Chilès, J.P.; Delfiner, P. Geostatistics: Modeling Spatial Uncertainty; Wiley: New York, NY, USA, 2012; p. 699. [Google Scholar]
- Goovaerts, P. Geostatistics for Natural Resources Evaluation; Oxford University Press: New York, NY, USA, 1997; p. 480. [Google Scholar]
- Wackernagel, H. Multivariate Geostatistics: An Introduction with Applications; Springer: Berlin/Heidelberg, Germany, 2003; p. 388. [Google Scholar]
- Mariethoz, G.; Caers, J. Multiple-Point Geostatistics: Stochastic Modeling with Training Images; Wiley: New York, NY, USA, 2014; p. 376. [Google Scholar]
- Langlais, V.; Beucher, H.; Renard, D. In the shade of the truncated Gaussian simulation. In Proceedings of the Eighth International Geostatistics Congress, Santiago, Chile, 1–5 December 2008; Ortiz, J.M., Emery, X., Eds.; Gecamin Ltda: Santiago, Chile, 2008; pp. 799–808. [Google Scholar]
- Le Blévec, T.; Dubrule, O.; John, C.M.; Hampson, G.J. Modelling asymmetrical facies sucessions using pluri-Gaussian simulations. In Geostatistics Valencia 2016; Gómez-Hernández, J.J., Rodrigo-Ilarri, J., Rodrigo-Clavero, M.E., Cassiraga, E., Vargas-Guzmán, J.A., Eds.; Springer International Publishing AG: Cham, Switzerland, 2017; pp. 59–75. [Google Scholar]

**Figure 1.**Plan views of geological domains (

**A**,

**C**) and indicator variograms of domain D

_{1}(

**B**,

**D**). The regularity of the domain boundary is reflected in the first-order derivative of the indicator direct variogram at the origin: regular domain boundary (

**A**) associated with a finite variogram slope at the origin (

**B**); fractal domain boundary (

**C**) associated with an infinite variogram slope at the origin (

**D**).

**Figure 2.**Plan views of geological domains (

**A**,

**C**) and indicator variograms of domain D

_{1}(

**B**,

**D**). The smoothness of the domain boundary is reflected in the second-order derivative of the indicator direct variogram at the origin: domain boundary with vertices highlighted in red (

**A**) associated with a strictly concave variogram near the origin (

**B**); smooth domain boundary (

**C**) associated with a straight variogram near the origin (

**D**).

**Figure 3.**Plan views of geological domains (

**A**,

**C**,

**E**) and indicator cross-variograms of domains D

_{1}and D

_{2}(

**B**,

**D**,

**F**). The surface area of the contact between domains D

_{1}and D

_{2}is reflected in the first-order derivative of the indicator cross-variogram at the origin: no contact between domains (

**A**) associated with a zero cross-variogram slope at the origin (

**B**); existence of a contact with a small surface area (

**C**) associated with a small negative cross-variogram slope at the origin (

**D**); existence of a contact with a large surface area (

**E**) associated with a large negative cross-variogram slope at the origin (

**F**).

**Figure 4.**Plan views of geological domains (

**A**,

**C**,

**E**) and absolute values of the ratios of indicator cross-to-direct variograms $-{\gamma}_{1,2}/{\gamma}_{1}$ (red curves) (

**B**,

**D**,

**F**); the expected sill value (Equation (20)) is represented by a black solid line.

**Figure 5.**Interpreted rock type model in a cross-section under study. Sample drill hole data have been superimposed. Local coordinates are used to preserve the confidentiality of the data.

**Figure 6.**Indicator direct variogram of rock type 1 (andesite) along directions dipping 30° (blue) and −60° (red) with respect to the north direction, calculated from (

**A**) drill hole data, (

**B**) interpreted model. Calculations consider an angle tolerance of 20° around the target directions. The same lag values have been used for both variograms.

**Figure 7.**Indicator direct variogram of rock type 3 (sedimentary rocks) along directions dipping 30° (blue) and −60° (red) with respect to the north direction, calculated from (

**A**) drill hole data, (

**B**) interpreted model. Calculations consider an angle tolerance of 20° around the target directions. The same lag values have been used for both variograms.

**Figure 8.**Indicator cross-variogram between rock types 2 (other volcanic rocks) and 3 (sedimentary rocks) along directions dipping 30° (blue) and −60° (red) with respect to the north direction, calculated from (

**A**) drill hole data, (

**B**) interpreted model. Calculations consider an angle tolerance of 20° around the target directions. The same lag values have been used for both cross-variograms.

**Figure 9.**Indicator cross-variogram between rock types 4 (granodioritic porphyry) and 5 (monzonitic porphyry) along directions dipping 75° (blue) and −15° (red) with respect to the north direction, calculated from (

**A**) drill hole data, (

**B**) interpreted model. Calculations consider an angle tolerance of 20° around the target directions. The same lag values have been used for both cross-variograms.

**Figure 10.**Indicator cross-variogram between rock types 4 (granodioritic porphyry) and 6 (hydrothermal breccia) along directions dipping 75° (blue) and −15° (red) with respect to the north direction, calculated from (

**A**) drill hole data, (

**B**) interpreted model. Calculations consider an angle tolerance of 20° around the target directions. The same lag values have been used for both cross-variograms.

**Figure 11.**Absolute value of the ratio of indicator cross-to-direct variograms between rock types 2 (other volcanic rocks) and 3 (sedimentary rocks) along directions dipping 30° (blue) and −60° (red) with respect to the north direction, calculated from (

**A**) drill hole data, (

**B**) interpreted model. Calculations consider an angle tolerance of 20° around the target directions. The same lag values have been used for both variogram ratios. The expected sill value (Equation (20)) is represented by a black solid line.

**Figure 12.**Absolute value of the ratio of indicator cross-to-direct variograms between rock types 3 (sedimentary rocks) and 2 (other volcanic rocks) along directions dipping 30° (blue) and −60° (red) with respect to the north direction, calculated from (

**A**) drill hole data, (

**B**) interpreted model. Calculations consider an angle tolerance of 20° around the target directions. The same lag values have been used for both variogram ratios. The expected sill value (Equation (20)) is represented by a black solid line.

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## Share and Cite

**MDPI and ACS Style**

Maleki, M.; Emery, X.; Mery, N.
Indicator Variograms as an Aid for Geological Interpretation and Modeling of Ore Deposits. *Minerals* **2017**, *7*, 241.
https://doi.org/10.3390/min7120241

**AMA Style**

Maleki M, Emery X, Mery N.
Indicator Variograms as an Aid for Geological Interpretation and Modeling of Ore Deposits. *Minerals*. 2017; 7(12):241.
https://doi.org/10.3390/min7120241

**Chicago/Turabian Style**

Maleki, Mohammad, Xavier Emery, and Nadia Mery.
2017. "Indicator Variograms as an Aid for Geological Interpretation and Modeling of Ore Deposits" *Minerals* 7, no. 12: 241.
https://doi.org/10.3390/min7120241