# ARDIE: AR, Dialogue, and Eye Gaze Policies for Human-Robot Collaboration

Chelsea Zou, Kishan Chandan, Yan Ding, Shiqi Zhang

###### Abstract

Human-robot collaboration (HRC) has become increasingly relevant in
industrial, household, and commercial settings. However, the effectiveness of
such collaborations is highly dependent on the human and robots’ situational
awareness of the environment. Improving this awareness includes not only
aligning perceptions in a shared workspace, but also bidirectionally
communicating intent and visualizing different states of the environment to
enhance scene understanding. In this paper, we propose ARDIE (Augmented
Reality with Dialogue and Eye Gaze), a novel intelligent agent that leverages
multi-modal feedback cues to enhance HRC. Our system utilizes a decision
theoretic framework to formulate a joint policy that incorporates interactive
augmented reality (AR), natural language, and eye gaze to portray current and
future states of the environment. Through object-specific AR renders, the
human can visualize future object interactions to make adjustments as needed,
ultimately providing an interactive and efficient collaboration between humans
and robots.

##  I INTRODUCTION

The ability for robots to collaborate effectively with humans is highly
dependent on a mutual scene perception and shared situational understanding
[1]. This requires not only the human and robot to understand and interpret
each other’s intentions, but also convey important contextual information to
enhance scene understanding. This is typically challenging because humans and
robots prefer to communicate information differently [2], [3]. While humans
communicate knowledge using a variety of modalities such as speech, gaze, and
gestures [4], robots tend to rely on digital information like text-based
commands or visual images [5]. However, these modes of communication by
themselves may not always be sufficient or clear enough to convey intentions
and goals effectively. In order to understand the other party, a shared
situational awareness and perceptual understanding of the scene must hold
between the human and robot.

Augmented reality (AR) technologies have been developed for a wide range of
applications such as immersive gaming, architectural design, educational
training, and more [6]. Recently, AR has gained significant traction in human-
robot collaboration (HRC) domains for its capability to provide a shared
perceptual medium between humans and robots [7, 8]. By overlaying digital
information onto physical objects, the human can interact with a digital
augmented scene representation. Several studies have demonstrated the benefits
of AR through its capacity to model objects as interactive scene elements [9,
10, 11], and more recently, AR has also been utilized to produce
visualizations of internal states of robots, enabling humans to leverage the
information to adapt robot behaviors accordingly [2].

Figure 1: In an object stacking domain, the human specifies intent for the
desired objects through natural language and eye gaze. ARDIE integrates these
feedback modalities to display future states of the task in AR. When the human
confirms the visualizations, a UR5e robot arm proceeds to object manipulation.

Despite its advantages, current AR for HRC has several limitations. One
challenge is the communication gap between humans and robots. Individual means
of communication such as verbal cues may not be sufficient to effectively
convey goals. Therefore, additional channels that can also reveal intentions,
such as eye gaze, should be considered to improve bidirectional communication.
Moreover, most AR research in HRC focuses on visualizing the present
environment and robot status [8]. However, it is equally crucial for humans to
visualize future states of the environment in order to make current decisions.
Our proposed system, ARDIE (AR with Dialogue and Eye Gaze), addresses this by
providing real-time visual feedback of prospective outcomes to allow humans to
make necessary adjustments before proceeding onto the next step.

We use a decision theoretic framework to model a joint policy that
incorporates eye gaze data from a mixed reality (MR) headset, as well as
language to enable intuitive communication. While gaze is oftentimes noisy and
can only reference location or position, dialogue, on the other hand, can
ground visual cues through responsive speech such as question asking and
answering, which can further disambiguate between item references regardless
of physical space. These two modalities used together can provide more
effective communication.

Figure 2: In this demonstration, the human wants to stack a large green block,
a small red block, and a small blue block. For each block, the human conveys
their intentions and an AR visualization is produced for the next state of the
object sequence.

We formulate our problem under the scope of decision-making under uncertainty,
where the state of the environment is not directly observable but can be
inferred from a set of observations, and the actions of the agents can
influence state transition and observations. Our joint policy specifies how
the human and robot should act in each state of the environment based on the
multi-modal feedback cues. While gaze is cheap and can reference various
locations in space effectively, dialogue can further disambiguate implicit
cues through direct speech and question answering. In turn, the joint output
of our decision model provides optimal actions that can improve overall
situational awareness.

Figure 3: The human conveys intentions through gaze and language. The AR
interface renders gaze as observation input into the POMDP planner. The
dialogue system processes human speech through a natural language
understanding (NLU) unit, and similarly, feeds that observation into the POMDP
planner. The outputs of the planner are either a gaze request action, a future
state AR visualization, a question asking action, or a robot action.

In this paper, we present ARDIE, a novel agent that enhances HRC by leveraging
intuitive human feedback cues through a joint AR, dialogue, and eye gaze
policy. Our system incorporates visualizations of current and future states to
effectively convey intentions and improve decision-making by enhancing
situational awareness. By providing a visual understanding of task outcomes,
ARDIE enables users to adjust plans accordingly, leading to more effective
collaboration. Our work provides several contributions. The first is a novel
multi-modal agent system that takes in natural human feedback to convey future
state visualizations for HRC tasks. Our second contribution is the
implementation of this framework through an MR headset and a UR5e robot arm
for an interactive object manipulation task.

##  II Related Works

In this section, we discuss a few research areas that are relevant to our work
on ARDIE.

AR for Robotics: Several studies have leveraged the capabilities of AR for
robotics. In the past, researchers proposed a system that enables a mobile
robot to project AR visual cues onto the floor, indicating intended actions to
nearby humans [12]. There have also been proposed methods for visualizing
intended assistance in shared control of robotic systems through trajectory
overlays [13], as well as telerobotic systems that allow human operators to
remotely control a robot and receive visual feedback through head-mounted
displays [14]. Instead, we use an approach that leverages natural human
feedback cues to provide a more intuitive HRC experience while our joint
policy allows a more robust strategy to minimize ambiguity through the
grounding provided by multiple feedback channels.

Natural Human Feedback Cues for HRC: Researchers have attempted to overcome
the communication barrier between humans and robots by using natural language,
where incorporating dialogue grounding into HRC systems can promote mutual
understanding and clarification [15], [16]. Overt eye gaze is shown to reveal
human intent in directed attention tasks [17], and gaze tracking has been
previously integrated into robotic control systems [18], [19]. Despite these
successes, single feedback channels may not be sufficient to resolve
ambiguities occurring in noisy environments. For example, speech recognition
is prone to translation errors and gaze may not be helpful in cluttered
environments. ARDIE integrates both gaze and dialogue modalities with AR to
adapt to dynamic environments in HRC.

Mixed Reality for Human Robot Communication: Researchers found that in an item
disambiguation task, an MR interface to highlight tabletop items was most
engaging, but the use of an AR projector was the most accurate [20]. In
another approach, the authors utilize an MR approach for human-robot
communication by combining eye gaze, language, and pointing gestures to
highlight and disambiguate between tabletop objects [21]. Instead, we leverage
the benefits of both AR and human feedback cues for a human-robot
_collaboration_ task, in which future states are shown using AR objects that
possess physically plausible properties of the real object, such as structure,
solid body, and shape. This component enables tasks that can not only
distinguish between objects, but also tasks that require simulating object
interactions, such as object stacking.

Internal State Visualizations: Researchers have developed interfaces that
enable human workers to directly visualize internal states intentions of
robots through AR [8], [2]. However, in domains where human knowledge is
essential in guiding robot tasks, it is critical for the human to be able to
visualize not only the robot’s intentions but also their own intended outcomes
regarding the next decision step. ARDIE enables visualizations of human
internal states by projecting the future state based on their current
decisions. This allows humans to understand effects of their own actions to
modify behaviors accordingly.

##  III Background on POMDPs

A partially observable Markov decision process (POMDP) is a mathematical
framework to model decision making in uncertain environments. Specifically, it
is used in tasks where the agent only has partial information about the
environment, and therefore must reason about its beliefs over possible states
of the environment. The system is modeled as a set of states $S$, observations
$\Omega$, actions $A$, and rewards $R$, with the goal to learn an optimal
policy $\pi$ that maximizes the expected cumulative reward over time.

A POMDP is formalized as a tuple of $(S,A,\Omega,T,O,R,\gamma)$. At each time
step, the environment is in some ground state $s\in S$, the agent takes an
uncertain action $a\in A$ and receives an assigned reward $R(s,a)$. The system
transitions to a new state $s^{\prime}$, as modeled by the conditional
probability transition function $T(s,a,s^{\prime})$ = $p(s^{\prime}|s,a)$.
Next, an uncertain observation $z\in\Omega$ is received according to the
observation function $O(s^{\prime},a,z)$ = $p(z|s^{\prime},a)$. Finally a
discount factor $\gamma\in[0,1)$ is applied to ensure the reward is finite.
Solving a POMDP produces a policy $\pi:B\mapsto A$ that maps a belief $b\in B$
to an action $a\in A$ that will maximize long-term rewards. A value function
$V_{\pi}(b)$ that specifies the expected total reward of executing $\pi$
starting from $b$, can be approximately by a convex, piecewise-linear
function:

$V(b)=\max\limits_{\alpha\in\Gamma}(\alpha\cdot b)$

where $\Gamma$ is a finite set of vectors called $\alpha$-vectors, each
associated with an action, $b$ is the discrete vector representation of a
belief, and $\alpha\cdot b$ is the inner product of vectors $\alpha$-vector
and $b$. The policy can be denoted by a set of $\alpha$-vectors. The agent
begins with an initial belief $b_{0}$, and maintains a belief $b(s)$ about the
state which is a probability distribution over all the possible states. Using
Bayes rule, the belief $b(s)$ is updated when receiving observation $z$ after
taking action $a$:

$b^{\prime}(s^{\prime})$ = $\gamma O(z|s^{\prime},a)\displaystyle\sum_{s\in
S}T(s^{\prime}|s,a)b(s)$

##  IV System Implementation

Consider a simple object assembly task, where the human can select various
objects with different properties from different locations, and a robot can
help sequentially stack these objects in a target region. We model our task as
a POMDP, where inputs are the user’s intentions, inferred from eye gaze and
dialogue responses. The system then generates AR visualizations of future
states that align with human intent.

###  IV-A AR Interface

There are three major components to the AR interface. The renderer allows the
human to display intent through eye gaze data for the different scene
elements. When a gaze action is realized, the requester queries the human gaze
through an MR headset, Magic Leap 1, to gather information regarding the
approximate location of the physical object. Through the visualizer, when a
visual action is realized, AR object representations are projected into the
scene and modeled according to the real-world object by considering shape,
solid body, mass, and gravity.

###  IV-B Dialogue System

The dialogue system has two elements to allow the agent to strategically ask
human questions and obtain spoken feedback. The natural language understanding
(NLU) unit interprets human speech and utilizes a text-to-speech ROS package.
The question-asking unit uses Google Cloud’s speech-to-text API to allow the
agent to query the human about the intended object or ask a confirmatory
question.

###  IV-C POMDP Planner

We use the APPL offline POMDP planning software, based on the SARSOP algorithm
for solving discrete POMDPs [22]. Recall that a POMDP is modeled as a tuple
$(S,A,\Omega,T,O,R,\gamma)$. Our system specifies the following:

  * •

$S$: The state space is the set of objects that correspond to the humans
intentions, where $S=\\{s_{1},s_{2},...,s_{n}\\}$ for $n$ different intended
objects in the environment.

  * •

$A$: The set of actions correspond to different decisions the agent can take.
Let $A$ = {$a_{gaze},a_{ask},a_{project}$} where $a_{gaze}$ requests the human
to look towards the intended object, $a_{ask}$ initiates the SDS to query the
human about the object, and $a_{project}$ displays a visualization of the
future state of the augmented environment, directly into the human’s field of
view.

  * •

$\Omega$: The set of observations contains the partially observed states of
the human’s true intentions. Each observation $z\in\Omega$ corresponds to
either a positional or descriptive property of the desired object, indicated
by gaze and dialogue respectively.

  * •

$T$: The transition function $T(s,a,s^{\prime})$ is an identity function for
actions $a_{gaze}$ and $a_{ask}$ where the state remains unchanged. The
$a_{project}$ actions are deterministic reporting actions that lead state
transitions to terminate.

  * •

$O$: The observation function $O(s^{\prime},a,z)$ models the noise in overt
human gaze and spoken language. For $a_{gaze}$ and $a_{ask}$ we assume the
probability of getting a correct observation is 0.8.

  * •

$R$: The reward function assigns the agent a large positive value for
displaying the future state of the correct item, and penalizes the agent with
a large negative value for an incorrect visualization. The gaze action is
assigned a smaller negative value than the ask action.

  * •

$\gamma$: The discount factor is set to 0.99 and determines the relative
importance of future rewards compared to immediate rewards.

###  IV-D Demonstration

Consider a case where the human wants to visualize and stack three specific
blocks sequentially, with a large green block on the bottom, then a small red
block, and finally a small blue block on top. ARDIE begins by prompting the
human to look at the first desired block to collect eye gaze position.
Afterwards, ARDIE engages in a dialogue with the human to ask for the color.
The human responds with ”green”, and after sufficient confidence, an AR
representation of the desired object is projected onto a target location.
ARDIE then asks a confirmatory question. If the configuration of the green
block is confirmed, a robot action is sent to a UR5e arm to pick up and place
the block accordingly. If the visualization does not look as intended to the
human, the agent is re-instantiated to the last state, allowing the human to
select another set of objects more appropriate for their goals. A similar
process is done for the next small red block, and later the small blue block.

##  V Conclusion and Future Works

We aim to bridge the communication gap in HRC tasks through ARDIE, a novel
intelligent agent that utilizes a joint AR, dialogue, and eye gaze policy to
improve situational awareness. ARDIE enables humans and robots to intuitively
convey their internal state intentions and visualize current and future states
of the environment. Our multi-modal system provides a more natural mechanism
for humans and robots to communicate and make better informed decisions.

We realize a practical limitation of our system is the feasibility of scaling
up to more complex and dynamic environments. One avenue for future work is to
examine the physics of object interactions in greater depth to help model
outcomes with better precision and accuracy. Incorporating more advanced
physics engines can help humans and robots better understand the forces and
constraints involved in object manipulation, leading to physically plausible
behaviors in response to changes in the environment. Overall, we aim to
progress efforts in improving coordination, communication, and collaboration
between humans and robots.

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∎

11institutetext: E. Farahbakhsh 22institutetext: Department of Mining
Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran,
Iran
22email<EMAIL_ADDRESS>33institutetext: A. Hezarkhani
44institutetext: Department of Mining Engineering, Amirkabir University of
Technology (Tehran Polytechnic), Tehran, Iran
55institutetext: T. Eslamkish 66institutetext: Department of Mining
Engineering, Amirkabir University of Technology (Tehran Polytechnic), Tehran,
Iran
77institutetext: A. Bahroudi 88institutetext: School of Mining Engineering,
College of Engineering, University of Tehran, Tehran, Iran
99institutetext: R. Chandra 1010institutetext: School of Mathematics and
Statistics, University of New South Wales, Sydney, Australia

# Three-dimensional weights of evidence modeling of a deep-seated porphyry Cu
deposit

Ehsan Farahbakhsh  Ardeshir Hezarkhani  Taymour Eslamkish  Abbas Bahroudi
Rohitash Chandra

(Received: date / Accepted: date)

###### Abstract

Given the challenges in data acquisition and modeling at the stage of detailed
exploration, it is difficult to develop a prospectivity model, particularly
for disseminated ore deposits. Recently, the weights of evidence (WofE) method
has demonstrated a high efficiency for modeling such deposits. In this study,
we propose a framework for creating a three-dimensional (3D) weights of
evidence-based prospectivity model of the Nochoun porphyry Cu deposit in the
Urmia-Dokhtar magmatic arc of Iran. The input data include qualitative
geological and quantitative geochemical information obtained from boreholes
and field observations. We combine ordinary and fuzzy weights of evidence for
integrating qualitative and quantitative exploration criteria in a 3D space
constrained by a metallogenic model of the study area for identifying a deep-
seated ore body. Ordinary weights of evidence are determined for geological
data, including lithology, alteration, rock type, and structure. Moreover, we
determine the fuzzy weight of evidence for each class of continuous
geochemical models created based on Fe, Mo, and Zn concentration values
derived from boreholes. We integrate the input evidential models using WofE
and create two prospectivity models (i.e., posterior and studentized posterior
probability). We also determine anomalous voxels in the probability models
using concentration-volume fractal models and validate them using prediction-
volume plots. The modeling results indicate that the studentized posterior
probability model is more efficient in identifying voxels representing copper
mineralized rock volumes. We provide open source software for the proposed
framework which can be used for exploring deep-seated ore bodies in other
regions.

###### Keywords:

Three-dimensional prospectivity modeling Weights of evidence Uncertainty
Porphyry Cu Nochoun

††journal: Journal

## Highlights

  * •

Three-dimensional weights of evidence method is an efficient tool for modeling
deep-seated ore bodies;

  * •

Probability models are able to show the dispersion of target mineralization in
depth;

  * •

Studentized posterior probability model provides more reliable results
compared to posterior probability model.

##  1 Introduction

The increasing shortage of easily detectable, outcropping ore deposits has led
more and more mineral explorers to prospect for concealed or deep-seated ore
deposits, in particular for metals such as copper, which plays a crucial role
in modern society (Mudd et al., 2013; Mudd and Jowitt, 2018; Schodde, 2013).
With increasing depth, traditional exploration methods are progressively
becoming less efficient and/or more costly. Mineral prospectivity mapping has
been developed and applied for various types of ore deposits and at a variety
of scales ranging from continental to regional (Brown et al., 2000; Carranza
et al., 2005; Carranza and Sadeghi, 2010; Chen and Wu, 2017; Rodriguez-Galiano
et al., 2015; Xiong and Zuo, 2018; Zuo et al., 2011). Whilst most
prospectivity mapping methods can be categorized as either knowledge- or data-
driven approaches (Cheng and Agterberg, 1999; Manap et al., 2013; Porwal et
al., 2003), hybrid methods consider both data and expert knowledge (Sun et
al., 2019). The latter are typically used for identifying areas of high
potential for the discovery of ore deposits in two-dimensional environments
(Carranza et al., 2008; Carranza and Laborte, 2015; Knox-Robinson, 2000;
Porwal et al., 2010).
Deep-seated ore deposits usually show weak exploration signals on the ground
surface. Therefore, there is a need to develop two-dimensional (2D) mineral
prospectivity mapping methods in a three-dimensional (3D) space to benefit
them for in-depth exploration of mineral resources. 3D modeling, analysis and
visualization facilitate the perception of key spatial factors in
mineralization, ore genesis, and geologic evolution in addition to target
appraisal (Carranza, 2009; Li et al., 2018; Mao et al., 2019; Payne et al.,
2015; Zuo et al., 2016). The ability of 3D modeling in providing a reliable
spatial model is completely dependent on the quality of input datasets,
modeling techniques, expert knowledge, and the complexity of the local
geological setting (Fallara et al., 2006; Houlding, 1994; Jessell et al.,
2014; Lindsay et al., 2012; Liu et al., 2016). A comprehensive metallogenic
model such as those suggested for porphyry Cu deposits (Berger et al., 2008;
Lowell and Guilbert, 1970; Meng, 1997; Sillitoe, 2010), helps geometric
modeling and spatial analysis through enhancing the reliability of 3D models.
Several three-dimensional mineral prospectivity mapping methods have been
developed in recent years (Li et al., 2015; Mao et al., 2019; Nielsen et al.,
2019, 2015; Xiao et al., 2015; Yuan et al., 2014), which can be applied along
with other modeling methods, such as geostatistics for modeling drilling data
and detecting deep-seated ore deposits at both regional and local scales.
Regional-scale 3D mineral prospectivity modeling and quantitative assessment
is rarely feasible, because required public-domain datasets with consistent
coverage over large areas are not available (Xiao et al., 2015). One of the
advantages of 3D mineral prospectivity modeling over traditional geostatistics
is its ability of integrating different types of qualitative and quantitative
exploration data rather than being restricted to modeling the concentrations
of individual geochemical elements. Some information, such as geological
characteristics obtained from boreholes consume large amounts of time and
money, which are surprisingly less considered at the stage of detailed
exploration due to the lack of a specific framework for combining qualitative
data with more favorable quantitative data. The 3D mineral prospectivity
modeling methods are able to integrate such data and provide an efficient
model for optimizing the process of selecting new drilling locations and
planning the exploitation of an ore reserve.
In recent years, Bayesian modeling approaches have been applied to modeling
geological features (Olierook et al., 2019; Scalzo et al., 2019). The Bayesian
inference approach is able to provide a fully quantitative and informative 3D
prospectivity model and to fuse all available constraints in a
probabilistically rigorous fashion. The weights of evidence (WofE) method is
based on Bayes’ rule (Xiao et al., 2015) and has been effectively used for 2D
mineral prospectivity mapping of various types of ore deposits (e.g., Carranza
(2004); Kreuzer et al. (2015); Pazand and Hezarkhani (2014); Porwal et al.
(2010); Zeghouane et al. (2016)). The fuzzy WofE method developed by Cheng and
Agterberg (1999), prevents loss of information due to converting continuous
models into binary models. This method has a number of advantages compared to
other simpler or even more complicated data-driven methods. In general, the
weights of evidence represent the degree of correlation between a target
mineralization and a particular model or pattern created under specific
conditions known as an evidence (Agterberg et al., 1990; Bonham-Carter et al.,
1989; Carranza, 2004; Cheng and Agterberg, 1999; Yuan et al., 2014).
Using 3D mineral prospectivity modeling, geological, geochemical and
geophysical data can be integrated according to the known dispersion of
mineralization in a modeling space. The result of this process is a formulated
3D model which presents a quantitative assessment of the probability of
detecting a target mineralization based on exploration criteria. The
exploration criteria must include all the factors which control a specific
target mineralization in a study area (Yuan et al., 2014). They help to
provide the 3D evidential models which are later used as inputs to the
modeling process.
In this study, we extend and demonstrate the application of WofE method in a
3D space based on a proposed framework for modeling a porphyry copper (Cu)
deposit located in southeast Iran within a magmatic arc called Urmia-Dokhtar.
We use borehole data consisting of qualitative geological and quantitative
geochemical data along with surficial geological data for providing input 3D
evidential models. Based on the proposed framework, the ordinary and fuzzy
WofE methods are used for weighting voxels in binary geological and continuous
geochemical evidential models, respectively. We use the total variance
associated with each voxel to create a studentized posterior probability model
in addition to a posterior probability model. The concentration-volume fractal
models and prediction-volume plots are used to evaluate and validate our
models. We provide Python scripts as open-source software for implementing the
proposed framework in this study.

##  2 Geological setting

The Nochoun porphyry copper deposit is located within a magmatic arc
subdivision called Urmia-Dokhtar where extensive Tertiary to Plio-Quaternary
intrusive and extrusive units are exposed in a northwest–southeast trend (Fig.
1). In several studies, a subduction-related magmatic model is suggested for
the Urmia-Dokhtar magmatic arc, which is known to be a result of the closure
of the Neo-Tethys ocean between Arabian and Eurasian plates (Berberian and
Berberian, 1981; Omrani et al., 2008). In general, this magmatic arc involves
two major mineralization regions of Chahar Gonbad to the southeast and Sungun
to the northwest. The dominant type of mineralization is porphyry Cu which is
associated with Eocene, Pliocene and Quaternary plutonic bodies and volcanic
rocks. The major lithological units of the study area located in the Chahar
Gonbad region in southeast Iran, consist of volcanic and subvolcanic
complexes, and intrusive bodies. The volcanic rocks cover most of the study
area and consist of Eocene andesite, dacite, and rhyodacite associated with
tuff breccias (Abedi et al., 2014). As shown in Fig. 2, the intrusive bodies
include granite to diorite dispersed in the south to southwest of the study
area.

Figure 1: A portion of the Chahar Gonbad region located in southeast Iran
within the Urmia-Dokhtar magmatic arc. The study area is shown at the center
of the map in a black square.

Figure 2: Detailed geological and alteration map of the study area along with
a cross-section intersecting some of the mineral occurrences. The black
rectangle shows the target area.

##  3 Alteration and mineralization

The hydrothermal alteration and mineralization in the Nochoun porphyry copper
deposit strike NE–SW and center on a stock (Fig. 2). The early hydrothermal
alteration was dominantly potassic and propylitic, followed later by phyllic,
silica, and argillic alteration types. The potassic alteration is represented
by mineral assemblages developed pervasively as halos around veins in the
central parts of the study area. There is a relatively sharp boundary between
the propylitic and potassic alteration zones in deeper parts of the ore
deposit, but this contact is obscured by later phyllic alteration in shallow
levels. The propylitic alteration is pervasive and represented mainly by
chloritization of primary and secondary biotite and groundmass materials of
the rocks which are peripheral to the central potassic zone (Fig. 2). The
plagioclase minerals were replaced with epidote, but this alteration is less
pervasive and less intense compared to the chloritization. Feldspar minerals
are locally altered to clay minerals in shallow levels and the dominant
mineral is kaolinite accompanied by illite. Moreover, the entire rocks have
been altered to an assemblage of clay minerals, hematite, and quartz close to
the erosion surface which are soft and white. The shallow alteration is
interpreted to represent a supergene blanket over the ore deposit and the
alteration of feldspar to clay in depth may have the same origin. Also, the
latter may represent an argillic stage of the hypogene alteration.
The hypogene copper mineralization has disseminated or taken place in veinlets
during phyllic alteration and to a lesser extent potassic alteration. During
the potassic alteration, the copper was deposited as chalcopyrite and minor
bornite; later hypogene copper was deposited mainly as chalcopyrite. The rocks
are highly altered at the exposed surface of the ore deposit and the only
mineral which has survived alteration is quartz. Most of the sulfide minerals
have been leached, and copper has been concentrated in an underlying supergene
zone by downward percolating groundwater. In general, the mineralization in
the study area is the result of two geological processes including the
intrusion of a granodiorite body and the ascending of hydrothermal fluids. The
mineralization-bearing rocks hosted by the granodiorite body usually appear in
the veins with a thickness of 2–3 m, and they are mostly found in the marginal
sections of the host body or tuffaceous units. This type of mineralization is
considered non-economic due to the low grade. Different types of ore minerals
such as hematite, oligist, magnetite, and malachite with intense
silicification of the veins are observed in mineralization zones (Fig. 3).
Moreover, the polished sections provided using mineralized rocks, show the
presence of chalcopyrite, pyrite, and sphalerite (Fig. 4). There are a number
of stockworks exposed on the surface which are the result of hydrothermal
fluid interaction (Fig. 2). These stockworks are probably related to the
intrusion of dykes and quartz monzonite apophyses. In this type of
mineralization, malachite and azurite minerals are found on the surface along
with minor chalcopyrite as inclusions within quartz (Abedi et al., 2014).

Figure 3: a) Cu oxide minerals including malachite and azurite in fractures of
silicified tuff crystals; b) quartz veinlets with a stockwork texture. Figure
4: Polished sections showing the presence of a) pyrite, and b) iron oxide
crystals in mineralized rocks.

##  4 Materials and methods

###  4.1 Drilling data

We use qualitative geological and quantitative geochemical data obtained from
113 boreholes to create 3D geological and geochemical evidential models. The
size of each voxel in the 3D models according to the extent of the modeling
space is determined 10$\times$10$\times$10 meters (m). The geometrical
parameters of the modeling space can be found in Table 1. We inscribe the 3D
models in a polygon created by the convex hull algorithm based on the
coordinates of the borehole collars on the ground surface (Fig. 5). Moreover,
they are restricted to a super- and sub-face based on the elevation of the
borehole collars and the depth of each borehole to make sure there are
sufficient number of data points for interpolating throughout the modeling
space. The total number of voxels is approximately 500,000.

Table 1: Geometrical parameters of the 3D modeling space. Parameter | Value
---|---
North–South extent | 970 m
East–West extent | 1,740 m
Vertical extent | 890 m
Polygon area on the surface | 0.8841 km2

The geological data used for creating 3D geological evidential models involve
lithology, alteration, and rock type information. In addition to Cu
concentration values used for creating a primary model of the target ore body,
we use other geochemical data including the concentration values of key
elements for creating 3D geochemical evidential models. The 3D strip-logs of
the geological and geochemical data are shown in Figs. 6 and 7, respectively.

Figure 5: Borehole collars along with their identification labels shown on the
geological map of the study area enlarged from Fig. 2. The polygon drawn
around the borehole collars shows the boundaries of the modeling space.

Figure 6: 3D strip-logs of the geological data including a) lithology, b)
alteration, and c) rock type.

Figure 7: 3D strip-logs of the geochemical data including a) Cu, b) Fe, c) Mo,
and d) Zn concentration values. The size and color of cylinders shown along
the boreholes are proportionate to the intensity of concentration values.
Highly positive skewed Zn concentration values yields a poor illustration of
the relevant 3D strip-logs.

###  4.2 Structural data

The drilling data used in this study do not involve structural data such as
the position of faults in depth. Therefore, we extend surficial structural
data to depth for creating 3D fault surfaces. The available surficial data
include some information about the strike, dip, and dip direction of the
faults which have been provided by the fieldworks, and validated by remote
sensing data (Farahbakhsh et al., 2019b). The faults include those exposed on
the surface or covered by a thin layer of regolith known as concealed faults.
A 3D model of the fault surfaces called 3D ribbons is created using the
available data which is shown in Fig. 8. The 3D ribbons are then restricted to
the modeling space and converted into a 3D block model to be used as a
structural evidential model.

Figure 8: 3D ribbon model showing the fault surfaces which have been created
using surficial structural data.

###  4.3 3D modeling

####  4.3.1 Geological modeling

As shown in Fig. 5, there are a high number of boreholes in a small area which
makes it possible to create a precise 3D geological model using the drilling
data. In this study, we use a basic interpolation method called closest point
for interpolating qualitative geological data including lithology, alteration,
and rock type in a 3D space with the RockWorks software package (RockWorks17,
2019b). The value of a voxel node is set to be equal to the value of the
nearest data point, regardless of its distance from the point or the value of
its other neighbors. One of the advantages of this method is that the solid
model nodes will honor the control points. This method can be used for
modeling complex non-stratiform geology (e.g., multiple intrusions, impact
craters, karst, etc.) such as our study area. The 3D geological evidential
models are constrained to the geological map presented in Fig. 5. Moreover,
the lithological model is constrained to the cross-sections shown in Fig. 9
drawn based on the drilling data and geological knowledge.

Figure 9: Lithological cross-sections presented in a 3D view. The 3D
lithological evidential model is constrained to these sections, in addition to
the geological map.

####  4.3.2 Geochemical modeling

The concentration values of different elements have been determined along the
boreholes. However, the number of samples are different. We investigate the
bivariate correlation between different elements and Cu concentration values.
According to the results, iron, molybdenum, and zinc (Fe, Mo, and Zn) show a
high correlation and they involve the highest number of samples. These
elements are usually applied through prospecting for porphyry Cu deposits
(Farahbakhsh et al., 2019a; Xiao et al., 2014). Moreover, these elements are
known as the key indications of porphyry Cu mineralization in the study area.
We use the inverse-distance anisotropic modeling method which is one of the
different kinds of the inverse-distance algorithm for interpolating the
geochemical data in a 3D space. This method has been widely used for
interpolating the concentration values of different geochemical elements in
both two- and three-dimensional models, particularly in porphyry copper
deposits (e.g., Wang and Huang (2012); Zuo (2011)).
Using the inverse-distance in general, we assign a voxel node value based on
the weighted average of neighboring data points, and the value of each data
point is weighted according to the inverse of its distance from the voxel
node, taken to a power. The greater the value of the exponent, the less
influence distant control points will have on the assignment of the voxel node
value (RockWorks17, 2019a; Zeghouane et al., 2016). Using the inverse-distance
anisotropic method, we look for the closest control point in each 90-degree
sector around the node. In this study, the weighting exponent is set to 2,
experimentally. The directional search can improve the interpolation of voxel
values that lie between data point clusters, and can be useful for modeling
borehole-based data. The quadrant searching tends to connect the limits (highs
and lows) at the same elevation.

###  4.4 Weights of evidence modeling

The weights of evidence method is a well-known and robust data-driven
prospectivity modeling method based on the Bayesian theorem (Bonham-Carter,
1994). This method is used for estimating the posterior probability of
detecting an ore body under the assumption of conditional independence of
input evidential models (Xiao et al., 2015). This assumption is also known as
one of the weaknesses of this method (Joly et al., 2012). The WofE method can
work with a low number of training datasets compared to other data-driven or
machine learning techniques. Using this method, the prior belief of detecting
a specific type of mineralization in an area or space is updated in the light
of other evidence such as geological, geochemical or geophysical models. An
example of the prior belief is the primary model created by interpolating the
concentration values of target element obtained through boreholes. In this
study, we interpolate the Cu concentration values as well as other geochemical
elements, and then convert into a binary model for determining the approximate
target ore body. In general, the posterior probability of mineralization
(P(M)) after looking at evidence (E) is determined via the likelihood function
using Eq. 1 (Bonham-Carter, 1994).

$P(M|E)=P(M)\frac{P(E|M)}{P(E)}$ (1)

This method is simple from the computational view, and 2D WofE method is
readily implemented using GIS packages (e.g., ArcGIS Desktop (2019); QGIS
Development Team (2019)); however, implementing this method in a 3D space is
more complicated. In this study, we deal with binary or discrete geological
and continuous geochemical evidential models. Therefore, to minimize the loss
of information, we determine ordinary positive and negative weights for each
unit of the geological models and a fuzzy weight for each class of the
geochemical models. In binary models, the weights of evidence enable a user to
interpret positive and negative weights in geological terms intuitively. The
positive (W+) and negative (W-) weights are determined using Eqs. 2 and 3
(Bonham-Carter, 1994).

$W^{+}=ln\frac{P(E|M)}{P(E|\bar{M})}$ (2)

$W^{-}=ln\frac{P(\bar{E}|M)}{P(\bar{E}|\bar{M})}$ (3)

These weights indicate the spatial association between the voxels with and
without mineralization, and the presence and absence of anomalous voxels in
evidential models, respectively. For example, an evidential or predictor model
can be used to assess the contribution of a geological process in the
formation and prospectivity of a specific type of mineralization (Xiao et al.,
2015). Investigating the contrast (C) which is determined using Eq. 4, can
help through this assessment (Bonham-Carter, 1994).

$C=W^{+}-W^{-}$ (4)

High values of contrast show strong association of an evidential model with
the mineralization process. The contrast is used to calculate the fuzzy weight
of each class in continuous models. In this study, we classify geochemical
evidential models based on the percentile to minimize the effect of number of
voxels on the fuzzy weight. Each model has ten classes with an equal number of
voxels and a fuzzy membership function is created for each set of contrasts
for different continuous evidential models. The contrast values are
transformed to the fuzzy space ranging from 0 to 1 using a logistic function
(Yousefi and Nykänen, 2016) and called fuzzy contrast ($\mu_{E}(C)$). The
fuzzy weight ($W_{\mu_{E}(C)}$) for each class of continuous models is
determined using Eq. 5 (Cheng and Agterberg, 1999).

$W_{\mu_{E}(C)}=ln\frac{\mu_{E}(C)P(E|M)+(1-\mu_{E}(C))P(\bar{E}|M)}{\mu_{E}(C)P(E|\bar{M})+(1-\mu_{E}(C))P(\bar{E}|\bar{M})}$
(5)

The posterior probability model is the result of integrating input evidential
models, which is generated using Bayes’ equation in a log-linear form (Eq. 6)
with the assumption that conditional independence applies (Bonham-Carter,
1994; Bonham-Carter et al., 1989).

$P_{Pst}=\frac{O_{Pst}}{1+O_{Pst}}$ (6)

Where, PPst denotes the posterior probability and OPst is the posterior odds.
The posterior odds equal the exponent of the posterior logit which can be
determined by Eq. 7 (Bonham-Carter, 1994; Cheng and Agterberg, 1999).

$lnO(M|E_{1}E_{2}\ldots
E_{k})=lnO(M)+\sum_{i=1}^{m}W_{i}^{+/-}+\sum_{j=1}^{n}W_{\mu_{j}}$ (7)

Where, there are k evidential models including m discrete and n continuous
models. We use the aggregate of positive or negative weights in discrete
models and the fuzzy weights in continuous models, with the prior logit of
mineralization in the study area for calculating the posterior logit.
Using the WofE method enables the user to calculate the effects of uncertainty
on the weights, and uncertainty due to missing information. This leads to
producing an uncertainty quantified model which is propagated in the decision
making. The variances of the weights and contrast help to model the
uncertainty of the posterior probability due to uncertainty in the weights and
caused by lack of information. The variance of positive ($\sigma^{2}_{W^{+}}$)
and negative ($\sigma^{2}_{W^{-}}$) weights in discrete models are determined
as presented in Eqs. 8 and 9, respectively (Bonham-Carter, 1994).

$\sigma_{W^{+}}^{2}=\frac{1}{N(E\cap M)}+\frac{1}{N(E\cap\bar{M})}$ (8)

$\sigma_{W^{-}}^{2}=\frac{1}{N(\bar{E}\cap
M)}+\frac{1}{N(\bar{E}\cap\bar{M})}$ (9)

Where, N is the number of voxels and the variance resulting from the
membership function $\mu_{E}$ can be expressed as Eq. 10. It has to be noted
that we have assumed $P(E)+P(\bar{E})=1$.

$\sigma_{\mu_{E}}^{2}[P(M)]=\frac{2\mu_{E}(1-\mu_{E})}{P(\mu_{E})}\sigma^{2}[P(M)]$
(10)

Where, $P(\mu_{E})$ denotes the probability of the membership function and
determined using Eq. 11. Also, $\sigma^{2}[P(M)]$ is the variance of the prior
probability of mineralization as given by Eq. 12.

$P(\mu_{E})=\mu_{E}P(E)+(1-\mu_{E})P(\bar{E})$ (11)

$\sigma^{2}[P(M)]=\\{P(M|E)-P(M)\\}^{2}P(E)+\\{P(M|\bar{E})-P(M)\\}^{2}P(\bar{E})$
(12)

A useful measure is to calculate the studentized value of contrast (CSt), as a
measure of the uncertainty with which the contrast is known. We calculate the
studentized value as the ratio of contrast to its standard deviation S(C) as
shown in Eq. 13.

$C_{St}=\frac{C}{S(C)}$ (13)

The standard deviation of contrast is determined using Eq. 14.

$S(C)=\sqrt{\sigma_{W^{+}}^{2}+\sigma_{W^{-}}^{2}}$ (14)

A large studentized contrast implies that the contrast is large compared with
the standard deviation, then the contrast is more likely to be real. A
studentized value larger than 2, or even 1.5 is satisfactory (Bonham-Carter,
1994). Due to the assumptions required for a formal statistical test,
particularly the problem with the dependence of the standard deviation of
contrast on the units of measurement, it is best to use this ratio in a
relative, rather than an absolute sense (Bonham-Carter, 1994).
We use the variances of the weights to calculate the variance of the posterior
probability at each voxel, and to generate a studentized posterior probability
model. The square root of total variance ($\sigma^{2}_{T}$) at each voxel
equals the standard deviation. The ratio of the posterior probability to the
corresponding standard deviation is called the studentized posterior
probability (PSt) as shown in Eq. 15.

$P_{St}=\frac{P_{Pst}}{\sqrt{\sigma_{T}^{2}}}=\frac{P_{Pst}}{\sqrt{\sum\sigma_{W^{+}}^{2}+\sum\sigma_{W^{-}}^{2}+\sum\sigma_{\mu_{E}}^{2}}}$
(15)

According to this equation, the ratio of the posterior probability to the
square root of variances of positive, negative and fuzzy weights in both
discrete and continuous models equals the studentized posterior probability in
each voxel. The studentized posterior probability acts as a measure of the
relative certainty of the posterior probability. The voxels where the
studentized value falls below some threshold can be masked out, due to lack of
confidence in the results. In this study, the threshold is determined based on
the concentration-volume (C-V) fractal model (Afzal et al., 2011).
As described above, the three-dimensional WofE method used in this study is
similar to the traditional method applied for mineral prospectivity mapping
(Bonham-Carter, 1994; Carranza, 2004; Cheng and Agterberg, 1999), and pixels
are replaced with voxels. We propose a framework for implementing the method
in a 3D space and summarize the steps to be taken for 3D mineral potential
modeling using the WofE method in Fig. 10. A brief description of these steps
is presented in the following.

  * •

Acquiring data including required geological and geochemical data;

  * •

Creating a primary model of the target mineralization based on the Cu
concentration values obtained along the boreholes in different intervals;

  * •

Providing 3D geological and geochemical evidential models;

  * •

Calculating the weights of evidence and other necessary parameters such as the
standard deviation and variance of the weights for different geological units
and each class of geochemical models;

  * •

Selecting input evidential models for creating the posterior probability model
based on the contrast and studentized contrast;

  * •

Integrating selected evidential models and creating 3D posterior probability
and studentized posterior probability models;

  * •

Validating the results.

Figure 10: Methodology flowchart of this study for 3D mineral prospectivity
modeling of an ore deposit using the WofE method.

##  5 Results

###  5.1 3D evidential models

We use three different types of geological data including lithology,
alteration, and rock type for creating 3D geological evidential models
constrained to the geological map. As shown in Fig. 5, we considered two
cross-sections ($AA^{\prime}$ and $BB^{\prime}$) along the small and large
diameters of the study area to visually investigate the correlation between
the aforementioned data types and the Cu concentration. These cross-sections
along with the interpolated Cu concentration anomaly zones (greater than 0.2%)
are shown in Fig. 11.

Figure 11: Cross-sections $AA^{\prime}$ and $BB^{\prime}$ (Fig. 5) illustrated
using a) lithology, b) alteration, and c) rock type data obtained from the
boreholes. The Cu concentration anomaly zones are shown in the background and
outlined using the black lines.

As mentioned earlier, we used RockWorks software package (RockWorks17, 2019b)
for creating 3D solid models using different available geological data
obtained from the boreholes. As described in section 4.3.1, according to the
high density of the data points, a method called the closest point is used to
interpolate the geological data in the 3D space. The geological models are
constrained to the geological map (Fig. 5). Moreover, the lithology model is
constrained to the cross-sections shown in Fig. 8 for providing a more precise
and reliable model. The solid models created using lithology, alteration, and
rock type data are presented in Fig. 12.

Figure 12: Solid models created using a) lithology, b) alteration, and c) rock
type data. Each solid model is shown with a cutout.

The 3D ribbon model of the faults shown in Fig. 8 is converted into a block
model. The size of each voxel in this model is the same as other evidential
models. Moreover, we use two buffer zones with a radius of 25 and 50 m
surrounding the fault blocks to investigate the correlation of proximity to
the faults and Cu mineralization. An overall model of the fault blocks and
different buffer zones is presented in Fig. 13.

Figure 13: Block model of the fault surfaces surrounded by two buffer zones in
a radius of 25 and 50 m.

We use the Cu concentration values in different intervals through the
boreholes to create a primary 3D model of the target ore body concealed in
depth using the inverse-distance anisotropic interpolation method which is
later used as a training model. Moreover, we use Fe, Mo, and Zn concentration
values for creating other geochemical evidential models. We present the
descriptive statistics of the output models in Table 2 and the anomalous
voxels in Fig. 14. The threshold used in each model for discriminating between
the anomaly population and the background except for the Cu concentration,
equals the 95th percentile. According to the cutoff grade of Cu in the ore
deposits located in the neighborhood of our study area such as Sarcheshmeh
(Waterman and Hamilton, 1975), the threshold for creating a binary model of
the target ore body is considered 0.4%. Based on this threshold, the target
ore body occupies almost 3.5% of the total modeling space, which can also be
considered as the prior probability (Fig. 14a).

Table 2: Descriptive statistics of the interpolated geochemical models.

Element | Minimum | Maximum | Mean | Median | Standard Deviation | Skewness | Kurtosis
---|---|---|---|---|---|---|---
Cu (%) | 0 | 1.9466 | 0.1758 | 0.1559 | 0.1095 | 2.8513 | 20.3918
Fe (ppm) | 1046.9552 | 227389.6406 | 50080.683 | 50797.1602 | 15825.3686 | 0.2302 | 2.9143
Mo (ppm) | 0.0155 | 2613.1301 | 187.9939 | 165.018 | 141.2469 | 2.023 | 10.4016
Zn (ppm) | 0.0327 | 27061.6719 | 339.8742 | 156.7078 | 726.8055 | 13.0335 | 273.5633

Figure 14: Anomalous voxels of the 3D geochemical models created using the
concentration values of a) Cu, b) Fe, c) Mo, and d) Zn.

###  5.2 Prospectivity modeling

####  5.2.1 Posterior probability model

Based on the equations presented in section 4.4, the ordinary weights of
evidence for geological evidential models are determined (Table 3). We
consider every unit of lithology, alteration, and rock type data as a binary
model in order to determine the weights of evidence, contrast, and studentized
contrast. Those lithology and alteration units as well as different rock types
and structural models which are not found in this table show negative contrast
and are removed from the modeling process. One of the reasons can be the low
number of occupied voxels, and hence low number of common voxels with the
target ore body. More details can be found in the supplementary file. The
fuzzy weight along with other important parameters are determined for each
class of the geochemical evidential models. In Tables 4–6, we present the
results for Fe, Mo, and Zn models, respectively. It is noteworthy that the
classes with negative contrast are neglected during the modeling process.

Table 3: Variation of the ordinary weights of evidence and other important parameters with different lithology and alteration units. | W+ | W- | Contrast | Studentized Contrast
---|---|---|---|---
Lithology |  |  |  |
Quartzolite | 0.7046 | -0.0016 | 0.7062 | 5.0496
Alteration |  |  |  |
Calcitized | 0.121 | -0.0026 | 0.1236 | 2.3717
Carbonatized | 0.6429 | -0.0033 | 0.6463 | 6.8551
Epidotized | 2.2307 | -0.0001 | 2.2308 | 2.732
Potassic | 0.1105 | -0.1611 | 0.2716 | 17.061
Silicified | 0.9206 | -0.0118 | 0.9324 | 16.2444
Table 4: Variation of the fuzzy weights of evidence and other important
parameters with different classes of Fe concentration values.

Lower Limit | Upper Limit | W+ | W- | Contrast | Studentized Contrast | Fuzzy Contrast | Fuzzy Weight
---|---|---|---|---|---|---|---
Min | 30086.0527 | -1.5855 | 0.0872 | -1.6728 | -31.5801 | 0.01 | 0.0863
30086.0527 | 38639.6172 | -0.1371 | 0.0142 | -0.1513 | -5.5568 | 0.6648 | -0.0116
38639.6172 | 43641.5 | -0.0218 | 0.0024 | -0.0242 | -0.9349 | 0.7551 | -0.0037
43641.5 | 47356.9648 | 0.0021 | -0.0002 | 0.0024 | 0.0923 | 0.7717 | 0.0004
47356.9648 | 50729.9414 | 0.0033 | -0.0004 | 0.0037 | 0.1437 | 0.7725 | 0.0006
50729.9414 | 54003.6758 | 0.0134 | -0.0015 | 0.0149 | 0.5815 | 0.7793 | 0.0027
54003.6758 | 57515.7734 | -0.028 | 0.0031 | -0.031 | -1.1927 | 0.7507 | -0.0046
57515.7734 | 61537.293 | -0.2199 | 0.0219 | -0.2418 | -8.5735 | 0.5916 | -0.0086
61537.293 | 67457.7891 | -0.0614 | 0.0066 | -0.068 | -2.5801 | 0.7259 | -0.0085
67457.7891 | Max | 0.8286 | -0.1469 | 0.9755 | 51.4817 | 0.99 | 0.7728

Table 5: Variation of the fuzzy weights of evidence and other important
parameters with different classes of Mo concentration values.

Lower Limit | Upper Limit | W+ | W- | Contrast | Studentized Contrast | Fuzzy Contrast | Fuzzy Weight
---|---|---|---|---|---|---|---
Min | 37.036 | -1.3426 | 0.0811 | -1.4237 | -30.1982 | 0.1718 | 0.0634
37.036 | 74.6444 | -0.4013 | 0.0365 | -0.4378 | -14.3426 | 0.6106 | -0.0181
74.6444 | 108.0786 | -0.0999 | 0.0105 | -0.1104 | -4.1198 | 0.7543 | -0.0165
108.0786 | 137.7597 | -0.0856 | 0.0091 | -0.0947 | -3.556 | 0.7602 | -0.0148
137.7597 | 166.287 | -0.0341 | 0.0037 | -0.0378 | -1.4497 | 0.7808 | -0.0069
166.287 | 196.6305 | -0.0773 | 0.0082 | -0.0855 | -3.2213 | 0.7636 | -0.0137
196.6305 | 232.1037 | 0.1468 | -0.0176 | 0.1644 | 6.7928 | 0.8436 | 0.047
232.1037 | 280.1264 | 0.2393 | -0.0302 | 0.2695 | 11.548 | 0.87 | 0.093
280.1264 | 357.9801 | 0.244 | -0.0309 | 0.2749 | 11.801 | 0.8712 | 0.0958
357.9801 | Max | 0.528 | -0.0783 | 0.6063 | 28.9463 | 0.9303 | 0.3223

Table 6: Variation of the fuzzy weights of evidence and other important
parameters with different classes of Zn concentration values.

Lower Limit | Upper Limit | W+ | W- | Contrast | Studentized Contrast | Fuzzy Contrast | Fuzzy Weight
---|---|---|---|---|---|---|---
Min | 67.9985 | -0.9715 | 0.0684 | -1.0399 | -26.2799 | 0.3132 | 0.0359
67.9985 | 84.6815 | -0.474 | 0.0417 | -0.5157 | -16.358 | 0.572 | -0.0125
84.6815 | 102.5373 | 0.1285 | -0.0153 | 0.1438 | 5.8967 | 0.8379 | 0.0394
102.5373 | 125.5783 | 0.274 | -0.0352 | 0.3093 | 13.4312 | 0.8789 | 0.1138
125.5783 | 155.639 | 0.2312 | -0.029 | 0.2602 | 11.1136 | 0.8678 | 0.0884
155.639 | 198.6275 | 0.1243 | -0.0147 | 0.139 | 5.6916 | 0.8366 | 0.0378
198.6275 | 267.4829 | 0.0086 | -0.001 | 0.0096 | 0.375 | 0.797 | 0.002
267.4829 | 394.8134 | 0.086 | -0.01 | 0.096 | 3.8698 | 0.8242 | 0.0239
394.8134 | 700.7863 | 0.0092 | -0.001 | 0.0103 | 0.4007 | 0.7972 | 0.0021
700.7863 | Max | 0.07 | -0.0081 | 0.078 | 3.1242 | 0.8188 | 0.0186

Based on the results presented in Tables 4–6, we plot the variations of
contrast and fuzzy weight, and variance of contrast and studentized contrast
with increasing the concentration values of the investigated elements shown in
Fig. 15.

Figure 15: Variations of contrast and fuzzy weight with the concentration
values of a) Fe, b) Mo, and c) Zn. Variations of variance of contrast and
studentized contrast with the concentration values of d) Fe, e) Mo, and f) Zn.

All the evidential models presented in Tables 3–6 are integrated as described
in section 4.4, and a posterior probability model is the result. The C-V
fractal model is used for classification and determining a proper threshold
for separating anomalous voxels. According to the C-V chart presented in Fig.
16a, the voxels showing a posterior probability greater than 0.23 are
considered as the certain anomaly. Also, the voxels showing a value greater
than 0.15 and 0.09 can be considered as the probable and possible anomaly,
respectively. In Fig. 16b, the classified posterior probability model and the
voxels with values greater than 0.08 are shown which occupy nearly 3% of the
modeling space.

Figure 16: a) C-V chart of the posterior probability of Cu mineralization in
the modeling space. b) Classified anomalous voxels extracted based on the
thresholds determined by the C-V chart.

####  5.2.2 Studentized posterior probability model

The total variance of each voxel is given by summing the variances of the
ordinary and fuzzy weights of evidence, which is considered as an estimate of
the uncertainty associated with each voxel. Similar to the posterior
probability model, we use the C-V chart for classification and separating the
anomalous voxels. According to the relevant chart shown in Fig. 17a, the
voxels with values greater than 4.9 are considered as a certain anomaly. Those
voxels showing values greater than 3.8 and occupying nearly 3% of the modeling
space are shown in Fig. 17b.

Figure 17: a) C-V chart of the studentized posterior probability of Cu
mineralization in the modeling space. b) Classified anomalous voxels extracted
based on the thresholds determined by the C-V chart.

###  5.3 Validation

In 2D mineral prospectivity mapping, a prediction-area plot is used in order
to quantitatively validate the results obtained from prospectivity maps
(Yousefi and Carranza, 2015). We extend the application of this type of plots
to three-dimensional space and call them prediction-volume (P-V) plots. In P-V
plots, we show the cumulative percentage of predicted mineralization and the
corresponding cumulative occupied volume, with respect to the total volume
against the prospectivity values. Therefore, the prediction ability of a
prospectivity model and its ability to delimit the modeling space for further
exploration and drilling are evaluated in a scheme. The P-V plot shows a curve
of the percentage (prediction rate) of known mineralization and a curve of the
percentage of occupied volume corresponding to the classes of a prospectivity
model. When an intersection point of the two curves is at a higher place, it
portrays a small volume containing a large number of mineralization-bearing
voxels. The comparison of prediction rates in the P-V plots shown in Fig. 18,
indicates the importance of analyzing the predictability of the prospectivity
models. We compared two prospectivity models including the posterior and
studentized posterior probability models. It is noteworthy that for assigning
probabilistic values to both models in terms of prospecting for Cu
mineralization, and distribution of the voxel values between 0 and 1, they are
transformed to a fuzzy space using a linear function.

Figure 18: P-V plots for a) the posterior probability and b) studentized
posterior probability prospectivity models.

##  6 Discussion

Most of the study area is covered by alluvium, therefore the surface outcrop
does not allow the lithology and alteration assemblages to be identified as
detailed as the drill cores. In this study, we used two different types of
drilling data including qualitative geological and quantitative geochemical
data obtained from 113 boreholes through creating a 3D prospectivity model of
Cu mineralization. The geological data comprise three types of data including
lithology, alteration, and rock type. Among the lithology types, quartzolite
units (i.e., intrusive rocks with a quartz content ¿90%) show positive
contrast and studentized contrast (Table 3) which were used as an input binary
model to the modeling process. The quartzolite units which involve less than
1% of the modeling space show a high studentized contrast indicating high
correlation of this lithology type with Cu mineralization in the modeling
space. Based on Figs. 11a and 12a, most of the modeling space is occupied by
andesite which shows a contrast near zero and uncorrelated with Cu
mineralization. The granodiorite units are in the second place in terms of
volume percentage and show a negative contrast. According to Fig. 11a, high-
grade Cu concentration is associated with stockwork veins mainly hosted by
andesite which can be due to the high porosity and permeability of this
lithology type caused by fractures. These probable fractures provide a path
for hydrothermal fluids through ascending to the ground surface. In Fig. 11a,
the cross-section $B-B^{\prime}$ appears to show a vertical pipe hosted by
andesite and characterized by elevated Cu concentration where the section
$A-A^{\prime}$ crosses section $B-B^{\prime}$. On the other hand, granodiorite
units can be considered barren or poorly mineralized.
Among the different types of alteration in the study area, calcitized,
carbonatized, epidotized, potassic, and silicified units show positive
contrast and studentized contrast (Table 3). These units were used as input
binary models to the modeling process. According to the results, potassic and
silicified units show the highest studentized contrast values, respectively.
It is noteworthy that the contrast of silica alteration is much higher than
the potassic alteration. This is compatible with the metallogenic model of the
mineralization in the study area, because Cu mineralization is mostly observed
along the silicified veins. The carbonatized units show a relatively high
contrast, and according to the metallogenic model, malachite and azurite are
considered as the main ore minerals. The epidotized and calcitized units are
the last alteration units in terms of studentized contrast. The epidotized
units occupy a small portion of the modeling space and the results are not
reliable, although they show a high contrast value. The calcitized units show
a low contrast value and then, a weak association with Cu mineralization in
the modeling space.
According to Fig. 6c, the rocks located in the modeling space originate from
five different sources, and none of them showed a positive contrast.
Therefore, we did not use this type of geological data in the modeling
process. We used surficial structural data through creating a 3D model of the
fault surfaces due to lack of structural information in the drilling data. In
addition to the fault surfaces which were converted into the fault blocks, two
different buffer zones in a radius of 25 and 50 m were created surrounding the
fault blocks. All the structural models show a negative contrast and were
removed from the modeling process. This confirms the field observations
implying that the Cu-bearing mineralization zones in the study area, are
mainly associated with azurite and malachite stockwork veins and less
associated with the faults.
Besides the qualitative geological data, we used the quantitative geochemical
data consisting of the concentration values of three elements including Fe,
Mo, and Zn. These elements are known as the key indicators of Cu
mineralization in the study area. The interpolated model of the Cu
concentration values used as the training model, ranges from 0 to nearly 2%
with an average of 0.17%. The model is positively skewed and shows a high
kurtosis value. Based on the descriptive statistics presented in Table 2, the
evidential model of Fe concentration is the only model which shows a normal
behavior with a skewness near zero and a kurtosis near 3 (Montgomery et al.,
2010). The concentration values in this model range from nearly 0.1 to more
than 22% with an average of 5%. The Mo model ranges from nearly 0.01 to more
than 2600 ppm, showing an average of nearly 188 ppm. This model is positively
skewed and shows a high kurtosis value. The Zn model which ranges from nearly
0.03 to more than 27000 ppm, is highly skewed and shows a very high kurtosis
value. As explained in section 4.4, to minimize the loss of information
through converting continuous models into binary models, we determined the
fuzzy weight of each class. Instead of using only a specific class of the
continuous models, we used all those classes showing a positive contrast in
the prospectivity modeling process. According to Tables 4–6, the highest
studentized contrast values belong to the classes from Fe (51.4817), Mo
(28.9463), and Zn (13.4312) models, respectively. Based on the graphs
presented in Fig. 15, there is no linear relationship between contrast and
fuzzy weight in continuous models, and a high contrast does not necessarily
yield a high fuzzy weight. On the other hand, we see a clear inverse
relationship between variance of contrast and studentized contrast.
We created two probability models including posterior and studentized
posterior probability models for visualizing Cu mineralization potential in
the target modeling space. It is clear that the studentized posterior
probability model is more reliable due to the contribution of uncertainty
through determining the anomalous voxels. According to Fig. 17b, where there
are a low number of boreholes, the total variance or uncertainty shows a high
value which gives a low studentized posterior probability. The intersection
point in the P-V plot of the posterior probability model shows 56% of the
anomalous voxels in the interpolated model of Cu concentration have been
predicted in 44% of the total modeling space. The intersection point in the
P-V plot corresponding to the studentized posterior model shows 59% of the
anomalous voxels in the interpolated model of Cu concentration have been
predicted in 41% of the total modeling space. The contribution of uncertainty
in creating the studentized posterior probability model leads to a higher
prediction rate compared to the posterior probability model.
According to Fig. 19, the ore body is divided into two parts. One is located
in a shallow depth and southwest of the modeling space, while the other is
located in a deeper level and northeast of the study area. The ore body
located in the northeast is restricted to the fault surfaces, implying the
younger age of the fault surfaces than the mineralization. The fault surfaces
in the southwest are related to the mineralization and surrounded by anomalous
voxels. These faults can be considered as the conduits through which the
hydrothermal fluids have risen up near to the ground surface. The ore bodies
and the difference in the depth of Cu mineralization are obvious and outlined
in Fig. 20.

Figure 19: Fault surfaces shown with the anomalous voxels of the studentized
posterior probability model (Fig. 17b). Figure 20: Side views of the
studentized posterior probability model to the a) north, and b) west.

We implemented the three-dimensional WofE modeling using Python scripts and
release them as open-source software. The main advantage of these scripts is
that they are simple and easy to understand for a user with basic knowledge of
programming and can be extended for other types of problems.

##  7 Conclusions

In this study, we created a mineral prospectivity model based on 3D weights of
evidence by integration of qualitative geological and quantitative geochemical
borehole data located on a porphyry Cu deposit in southeast Iran. To minimize
the loss of information, we determined the ordinary and fuzzy weights of
evidence for discrete and continuous evidential models, respectively. The
integration of various input evidential models provided two prospectivity
models including posterior probability and studentized posterior probability
models. The C-V fractal models were used through finding suitable thresholds
for classifying and separating anomaly populations from the background.
Although the studentized posterior probability model is more reliable due to
the lower uncertainty, both the posterior and studentized posterior
probability models show an acceptable prediction rate based on the P-V plots.
The results show the efficiency of our framework in constructing different
geometric models of a specific ore deposit concealed in depth. The proposed
framework helps in determining key factors which control the mineralization in
the modeling space, identifying potential mineralization, and improving the
perception of ore genesis. Among the lithological units, quartzolite shows a
high correlation with Cu mineralization in the modeling space. Moreover,
potassic, silica, carbonate, and propylitic alteration types are strongly
associated with Cu mineralization. The fuzzy weights determined for continuous
models show a high correlation between Cu and Fe concentration values, while
Mo and Zn show an average correlation. According to the results of mineral
prospectivity modeling, it is concluded that Cu mineralization is mainly
associated with stockwork veins which confirms the field observations. The Cu
mineralization is identified in two separate bodies located in different
depths covered by a thin layer of regolith. The hybrid application of ordinary
and fuzzy weights of evidence yields a promising result for the future
studies. The 3D mineral prospectivity modeling based on the weights of
evidence is highly dependent on the primary model created using the
concentration values of a specific target element. In future, we aim to use
geostatistical methods for creating a more reliable model to consider as the
primary target ore body, since we can provide an uncertainty model beside the
model yielded by the interpolation process. In addition, we can decrease the
uncertainty and increase the prediction rate in our posterior probability
model by adding other exploration data such as geochemical and geophysical
models and creating more input evidential models compatible to the
metallogenic model of the study area.

## Supplementary data

The Python scripts for implementing the proposed framework and supplementary
files including more details on the evidential models are available at
https://github.com/intelligent-exploration/3D_WofE.

###### Acknowledgements.

We acknowledge National Iranian Copper Industries Company for providing the
data.

## Conflict of interest

The authors declare that they have no conflict of interest.

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