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minerals Article

Kriging Interpolation for Evaluating the Mineral Resources of Cobalt-Rich Crusts on Magellan Seamounts Dewen Du 1,2, * 1

2

*

ID

, Shijuan Yan 1,2 , Fengli Yang 1 , Zhiwei Zhu 1 , Qinglei Song 1 and Gang Yang 1

Key Laboratory of Marine Sedimentology and Environmental Geology, First Institute of Oceanography, SOA (FIO), Qingdao 266061, China; [email protected] (S.Y.); [email protected] (F.Y.); [email protected] (Z.Z.); [email protected] (Q.S.); [email protected] (G.Y.) Evaluation and Detection Technology Laboratory of Marine Mineral Resources, National Laboratory for Marine Science and Technology, Qingdao 266061, China Correspondence: [email protected]





Received: 9 July 2018; Accepted: 27 August 2018; Published: 29 August 2018

Abstract: The evaluation of mineral resources on seamounts by geostatistics faces two key challenges. First, the conventional distance/orientation- and the simple distance-based variogram functions used are ineffective at expressing the spatial self-correlation and continuity of cobalt-rich crust thicknesses on seamounts. Second, the sampling stations used for a single seamount are generally very sparsely distributed because of the high survey costs, which results in an insufficient number of information points for variogram fitting. Here, we present an alternative geostatistical method that uses distance/gradient- and distance/relative-depth-based variograms to process data collected from several neighboring seamounts, allowing the variogram fitting. The application example reported for the Magellan seamounts demonstrates the suitability of the method for evaluating the mineral resources of cobalt-rich crusts. The method could be effective also for the analysis of surface data obtained from mountain slopes on land (e.g., soil). Keywords: seamount; mineral resource evaluation; cobalt-rich crusts; geostatistics

1. Introduction Seamounts are submarine volcanogenic conical or flat-topped mountains (the latter are called guyots) rising hundreds or thousands of meters from the seafloor, sometimes emerging above the sea level to form islands, and covering large areas (up to 10,000 km2 each). Cobalt-rich crusts form on the sediment-free surfaces of seamount slopes and summits, being of significant economic interest because of the potential presence of manganese, cobalt, nickel, tellurium, platinum, and other rare earth elements [1–3]. Therefore, several explorative and resource evaluation campaigns are currently conducted on seamounts. Since the distribution of sampling stations on seamounts tends to be sparse owing to the costs of the survey cruises, the spatial interpolation of the collected data represents a critical issue for both scientific research and industrial exploration. Window averaging, distance inverse weighting, simple average values, and geostatistics have been generally used for spatial interpolation. Among the interpolation methods, kriging is nowadays a typical choice for quantifying mineral reserves because it provides the best linear unbiased predictions [4–6]. Distribution of cobalt-rich crusts on seamount surfaces are influenced by the water depth [1,7,8] but, to the best of our knowledge, no study has proven that they are influenced by the direction. Kriging with conventional variograms [6,9–16], including orientation-based ones, has proven ineffective at

Minerals 2018, 8, 374; doi:10.3390/min8090374

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quantifying mineral resources on seamounts [17]. This is probably why kriging is not among the methods commonly used seamount interpolation mentioned above [18]. Minerals 2018, 8, x FOR PEERfor REVIEW 2 of 15 A new method using distance/gradient-based variograms has been recently proposed by at is quantifying mineral on seamounts [17]. This mineral is probably why kriging is not its Du etineffective al. [17] and preliminarily usedresources for the evaluation of seamount resources [19]. Despite among the methods commonly used for seamount interpolation mentioned above [18]. demonstrated efficiency in spatial interpolation for seamounts [17], some significant problems arise A new method using distance/gradient-based variograms has been recently proposed by Du et when using it. First, kriging interpolation must face the expensiveness of survey cruises for deep al. [17] and is preliminarily used for the evaluation of seamount mineral resources [19]. Despite its sea exploration and the consequent distribution of the [17], sampling stations, which results demonstrated efficiency in spatial sparse interpolation for seamounts some significant problems arisein an insufficient number of information points for variogram fitting, the solution for this problem was not when using it. First, kriging interpolation must face the expensiveness of survey cruises for deep sea presented in Duand et al., [17]. Second, fitting ofofdistance/gradient-based variograms exploration the2017 consequent sparse the distribution the sampling stations, which results in is anmore insufficient number information points variogram fitting, thewidely. solution for this problem was not complicated than that ofofconventional onesfor and is difficult to use presented in Du al., to 2017 [17]. Second, of interpolation distance/gradient-based variograms more In this study, weetaim promote the usethe offitting kriging for seamounts. First,isthe method complicated than that of conventional ones and is difficult to use widely. for unifying data collected from multiple seamounts to fit variograms for kriging interpolation on one In this study, we aim promote the use of kriging interpolation seamounts. First, guyot is described. Next, thetodistance/gradient-based variogram isforreintroduced, andthe anmethod alternative for unifying data collected from multiple seamounts to fit variograms for kriging interpolation on one (i.e., the distance/relative-depth-based variogram), which is easier for experimentation and one guyot is described. Next, the distance/gradient-based variogram is reintroduced, and an variogram fitting and is similarly effective, is presented for the first time. Then, the processing of alternative one (i.e., the distance/relative-depth-based variogram), which is easier for experimentation surveying data for fitting severaland guyots from the Magellan seamounts with above is discussed. and variogram is similarly effective, is presented for the firstthe time. Then,methods the processing of Last, surveying the kriging interpolation is compared with several non-geostatistical interpolation methods in data for several guyots from the Magellan seamounts with the above methods is discussed. orderLast, to demonstrate its effectiveness for the with application to seamounts. interpolation methods in the kriging interpolation is compared several non-geostatistical order to demonstrate its effectiveness for the application to seamounts.

2. Data 2. Data

Since the 1990s, the Magellan seamounts, which are located in the Western Pacific Ocean (Figure 1), the 1990s, the Magellan seamounts, located in the Western Pacific Ocean (Figure 1), have beenSince investigated by geologists from thewhich Chinaare Ocean Mineral Resources Association (COMRA) have been investigated by geologists from the China Ocean Mineral Resources Association (COMRA) as promising cobalt-rich crust deposits. Hence, we selected four guyots (Me, Mk, Ma, and Mc) from as promising cobalt-rich crust deposits. Hence, we selected four guyots (Me, Mk, Ma, and Mc) from the Magellan seamounts to serve as application examples of the geostatistical methods that have been the Magellan seamounts to serve as application examples of the geostatistical methods that have been described above. described above.

Figure 1. The selected guyots on the Magellan seamounts. Each guyot has an area of approximately

Figure 1. km The selected on thelocations Magellan Eacheach guyot has area of approximately 2 to 2; the sampling 3000 4000 kmguyots are seamounts. located away from other byan approximately 1 km to 2 2 3000 km to 4000 km ; the sampling locations are located away from each other by approximately 1 km 10 km. to 10 km. The bathymetric data for these guyots were obtained using a Simrad EM120 multibeam echo sonar (Kongsberg Underwater Technology Inc., Lynnwood, WA, USA) with a grid size of 100 × 100 m. The

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The bathymetric data for these guyots were obtained using a Simrad EM120 multibeam echo sonar (Kongsberg Underwater Technology Inc., Lynnwood, WA, USA) with a grid size of 100 × 100 m. The data for the cobalt-rich crust thicknesses were obtained by geological sampling, primarily via submarine drilling and dredging. The total number of sampling stations used for these four guyots was 264, and the water depth from sea level ranged from 1350 m to 3600 m. The spatial distributions of the cobalt-rich crusts were observed to be similar among the four seamounts [18]. Guyot Mk was used as the primary seamount that was to be interpolated via kriging, whereas the remaining three were subsequently added to the experiment to assess the accuracy with which they fitted the resulting variogram. Table 1 presents the statistical parameters of the cobalt-rich crusts. The significant difference between the average crust thicknesses of the four guyots indicates that it is unreasonable to simply combine (i.e., average) their data. Table 1. Original and unified parameters of the cobalt-rich crust thickness of the four selected guyots. Guyots

Statistic Parameters Original mean (mm) Transformed mean (mm) Original standard deviation (mm) Transformed standard deviation (mm) Transformed weighting (Ti ) Number of samples

Mk

Me

Ma

Mc

58 58 33 33 1.00 77

66 58 35 31 0.88 73

74 58 38 30 0.79 69

122 58 73 35 0.48 45

Unified Data 58 32 264

3. Materials and Methods In this section, we present two key aspects of the proposed geostatistical method. First, we describe two adapted distance/depth-based variograms: distance/gradient- and distance/relative-depth-based variograms. Then, we unify data collected from multiple seamounts to obtain a variogram to use for the kriging interpolation of cobalt-rich crusts on a seamount. 3.1. Distance/Depth-Based Variograms A perfectly conical seamount could be represented by a Lambert conformal coordinate system, but a local coordinate system can always be defined for any more complex surface, such as the guyots of the Magellan seamounts. Based on such a local coordinate system, one can build a conformal grid where the variogram is calculated and the kriging interpolation is performed. This operation consists in computing the variogram in a Riemannian space instead of a Euclidean one. This method has been investigated since 1998 [20] and is still used in the oil and gas field for estimating the variogram along non-flat faulted layers [21,22]. In a similar way, kriging methods for evaluating cobalt-rich crusts on seamounts are presented as follows. 3.1.1. Distance/Gradient-Based Variogram By defining a local coordinate system u = {u(x, y, dp)} on the surface of seamounts, where (x, y) and dp are, respectively, the geographical coordinates and the depth of a spatial variable, such as the thickness of cobalt-rich crusts notated as Z(u), the variogram equation can be written as γ(h) = 12 E([Z(u + h) − Z(u)]2 ), where h denotes the distance of geographical coordinates or differences of depth between two points in the coordinate system u. Since Z (u) = Z ( xi , yi , dpi ), i = 1, . . . , n represents the thickness data for cobalt-rich crusts on a seamount surface at a geographic position ( xt , yi ) with a water depth dpt , the distance/gradient-based

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variogram is defined as half the arithmetic mean of the variance of data pairs given by the distance/gradient [17], which can be expressed as: N (dis,g)

1 γ(h) = γ(dis, g) = × [ Z ( xi , yi , dpi ) − Z ( x j , y j , dp j )] ∑ 2N (dis, g) i=1,...,n,j =1...,n

2

(1)

where dis = sqrt[( xi − x j )2 + (yi − y j )2 ], g = arctan( dpi − dp j /dis) is the gradient of the seamount slope, and N (dis, g) is the number of point pairs in the group (dis, g). 3.1.2. Distance/Relative-Depth-Based Variogram Here, we define the distance/relative-depth-based variogram as half the arithmetic mean of the variance of data pairs given by the distance and relative depth, which can be expressed as: N (dis,∆dp)

1 γ(dis, ∆dp) = × [ Z ( xi , yi , dpi ) − Z ( x j , y j , dp j )] ∑ 2N (dis, ∆dp) i=1,...,n,j =1...,n

2

(2)

where ∆dp = dpi − dp j . 3.2. Unifying Data from Multiple Seamounts As noted above, the sparse distribution of sampling stations owing to the high survey costs results in an insufficient number of information points. Since such so-obtained calculation results are not ideal, for our application test, we selected several seamounts adjacent to each other with similar distribution characteristics in terms of cobalt-rich crust thickness, that is, the guyots Mk, Me, Ma, and Mc on the Magellan seamounts (Figure 1). Then, we unified the data collected from each of these seamounts, for experiments and variogram fitting using the kriging interpolation method, as follows: First, we divided the average value of the data sampled from one seamount (e.g., guyot Mk) to be interpolated by kriging by the average value of those collected from another one (i.e., guyot Me, Ma, or Mc). The resulting quotient, expressed as Ti = mean Mk /meani , i = Mk, Me, Ma, Mc, served as a weighting coefficient to transform the unified data. By supposing that the unified data are Zi,j (i = Mk, Me, Ma, Mc; j = 1, 2, . . . , Ni ), where Ni is the number of data points for a seamount i, those from the selected seamounts can be described as: ∗ Zi,j = Ti × Zi,j (i = Mk, Me, Ma, Mc; j = 1, . . . , Ni )

(3)

4. Results 4.1. Unified Data from Four Guyots on the Magellan Seamounts The datasets (i.e., cobalt-rich crusts thickness) from different seamounts exhibit different mean and standard deviation values (shown in Table 1) probably because of their different geological ages and environments; furthermore, it seems that we cannot directly use all the data together because of the non-stationarity of the data between different mounts. We attempt to transform the datasets from different mounts and form new datasets that exhibit similar mean and standard deviation values; furthermore, we combine these datasets to obtain stationary unified datasets. Using Equation (3), we obtained 264 unified data points from the statistical parameters shown in Table 1, and they had the same average values and approximate average deviations. Statistical tests revealed that the unified data followed a natural logarithm normal distribution, as shown in Figure 2. Thus, after logarithmic transformation, they were considered as weakly stationary random variables for variogram experiments [5]. Hence, this set of 264 data points, once transformed into natural

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logarithm values, can be used for experiments and variogram fitting of cobalt-rich crust thicknesses on the Magellan seamounts. Minerals 2018, 8, x FOR PEER REVIEW

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Figure 2. Logarithmic normal distribution histogram of the unified and transformed data.

Figure 2. Logarithmic normal distribution histogram of the unified and transformed data. 4.2. Experimental Variogram

4.2. Experimental Variogram

4.2.1. Distance/Gradient-Based Variogram Figure 2. Logarithmic normal distribution histogram of the unified and transformed data.

4.2.1. Distance/Gradient-Based For the distance search, weVariogram followed the approach described by Deutsch and Journel [23], while we 4.2. Experimental Variogram

divided the gradients intowe three groupsthe on approach the basis of the one reported by Du al. [17].[23], Pairs of we For the distance search, followed described by Deutsch andetJournel while sampling stations within a specific lag distance were selected according to both the lag distance and 4.2.1. Distance/Gradient-Based Variogram divided the gradients into three groups on the basis of the one reported by Du et al. [17]. Pairs of the tolerance. Then, the eventual statistical lag distance was taken as the mean of each group. Due to For thewithin distanceasearch, we followed the approach Deutsch andto Journel we sampling stations specific lag distance weredescribed selectedbyaccording both[23], thewhile lag distance and the same frequencies of point pairs occurring within certain gradient intervals, all point pairs were divided the gradients into three groups on thedistance basis of the one reported by Du et al.of [17]. Pairs of the tolerance. Then, the eventual statistical lag was taken as the mean each group. Due to divided into three groups. Hence, for lag dis distance and g, we obtained pairs andtothe variogram was calculated sampling stations within a specific were selectedNaccording both the lag distance and the same ofHere, point pairs within gradient intervals, all pointofpairs usingfrequencies Equation (1). and occurring gstatistical indicatelagthe meancertain distance and respectively, each were the tolerance. Then, the dis eventual distance was taken as the gradient, mean of each group. Due to divided into three groups. Hence, and g, we obtained N pairs and theallvariogram group. a result, the variogram fordis the three gradient groups was derived. theAs same frequencies of pointfor pairs occurring within certain gradient intervals, point pairs was were calculated divided(1). threethe groups. Hence, for dis and we obtained N pairs and the variogram was calculated Figure 3into shows variogram of cobalt-rich crust thicknesses thegroup. using Equation Here, disdistance/gradient-based and g indicate theg,mean distance and gradient, respectively, of for each using Equation (1). Here, dis and g indicate the mean distance and gradient, respectively, of each unified data, demonstrating its effectiveness at describing the self-correlation of mineral resources on As a result, the variogram for the three gradient groups was derived. group. As a result, thedistance/orientation-based variogram for the three gradient groups was derived. seamounts, whereas the variogram shown in Figure 4 seems ineffective. Figure 3 Figure shows3 the distance/gradient-based variogram of cobalt-rich crust thicknesses for the shows the distance/gradient-based variogram of cobalt-rich crust thicknesses for the unified data, demonstrating its effectiveness self-correlation of mineral on unified data, demonstrating its effectivenessat at describing describing thethe self-correlation of mineral resources resources on seamounts, the distance/orientation-based variogram shown in Figure 4 seems 4ineffective. seamounts, whereaswhereas the distance/orientation-based variogram shown in Figure seems ineffective.

(a) Experimental variograms for 1.2° gradient group (a) Experimental variograms for 1.2° gradient group

Figure 3. Cont.

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(b) Experimental variograms for 3.0° gradient group

(b) Experimental variograms for 3.0° gradient group

(c) Experimental variograms for 17.5° gradient group Figure 3. Experimental distance/gradient-based variograms for three gradient groups. The initial (c)scatter Experimental variograms for 17.5° gradient group distance-variation diagrams were fitted with artificial curves. Then, the mean square deviation Figure 3. Experimental distance/gradient-based variograms for three gradient groups. The initial C, the nugget effect C0, and the corresponding range were estimated according to the interpolated Figure 3. Experimental distance/gradient-based variograms for three gradient groups. The initial distance-variation variogramscatter curves. diagrams were fitted with artificial curves. Then, the mean square deviation

Variogram of thickness of crusts r(h)

variogram curves.

Variogram of thickness of crusts r(h)

distance-variation diagrams were fitted with artificial curves. Then, the mean square deviation C, the nugget effect Cscatter 0 , and the corresponding range were estimated according to the interpolated o o C, the nugget effect C0, and the corresponding range to the interpolated 90estimated 0o 45o were 135according variogram curves.

1.6

1.6

0o

45o

90o

135o

1.2

0.8

1.2 0.4

0.8

0.4

0 0

4

8

12

16

20

lag of distance(km)

Figure 4. Experimental distance/orientation-based variograms. 0°, 45°, 90°, and 135° (measured clockwise from north) 0 are the four orientations in which the point pairs are searched to calculate the variograms (by Du et al., 2017 [17]). 0 4 8 12 16 20

lag of distance(km) Figure 4. Experimental distance/orientation-based variograms. 0°, 45°, 90°, and 135° (measured

Figure 4. Experimental variograms. 0◦ ,are 45◦searched , 90◦ , and 135◦ (measured clockwise from north) distance/orientation-based are the four orientations in which the point pairs to calculate the clockwise from(by north) four orientations in which the point pairs are searched to calculate the variograms Du etare al.,the 2017 [17]). variograms (by Du et al., 2017 [17]).

Despite the different mean gradients and ranges between the three, the mean square deviations and the nugget effect were about 1.02 and 0.4, respectively, for all of them (Figure 3). Compared with

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the experimental distance/direction-based variogram [17], the algorithm Despite the different mean gradients and ranges between thedistance/gradient-based three, the mean square deviations clearly better applies to mineral resource data on seamount slopes. and the nugget effect were about 1.02 and 0.4, respectively, for all of them (Figure 3). Compared with the experimental distance/direction-based variogram [17], the distance/gradient-based algorithm

4.2.2.clearly Distance/Relative-Depth-Based Variogram better applies to mineral resource data on seamount slopes.

Our experiment on the distance/relative-depth-based variogram was similar to that on the 4.2.2. Distance/Relative-Depth-Based Variogram distance/gradient-based one. The key difference was the placement of the gradient groups at Our experiment on the similar to that on the (2). relative-depth intervals, such as distance/relative-depth-based 100 m, 200 m, 300 m, 400 m,variogram and 500 m,was as calculated in Equation distance/gradient-based one. The key difference was the placement of the gradient groups at relativeDistance-based variogram fittings were performed in each relative-depth interval for the conventional depth intervals, such as 100 m, 200 m, 300 m, 400 m, and 500 m, as calculated in Equation (2). Distancevariogram model. Intervals larger than 400 m exhibited no spatial correlation, and lower ones had based variogram fittings were performed in each relative-depth interval for the conventional insufficient data pairs, whereas the 400 m intervals (0–400 m, 400–800 m, and 800–1200 m) were suitable variogram model. Intervals larger than 400 m exhibited no spatial correlation, and lower ones had for the variogram calculation in thisthe study. insufficient data pairs, whereas 400 m intervals (0–400 m, 400–800 m, and 800–1200 m) were Figure shows the experimental distance-based suitable5for the variogram calculation in this study. variogram for the 0–400 m relative-depth interval, whereas the other variograms with the 400 m interval variogram (e.g., 400–800 m 0–400 and 800–1200 m) were all Figure 5 shows the experimental distance-based for the m relative-depth interval,Hence, whereas other variograms with the 400 m interval 400–800 m andinterval 800–1200ranging m) were from ineffective. inthe such situations, only the variogram with(e.g., a relative-depth all400 ineffective. Hence, situations, only the variogram with a relative-depth interval ranging 0 m to m can be used in forsuch kriging interpolation. from 0 m to 400 m can be used for kriging The fitting and kriging processes for theinterpolation. distance/relative-depth-based variogram were the same The fitting and kriging processes for the distance/relative-depth-based variogram were the same as for conventional variograms. The following section shows how the two distance/depth-based as for conventional variograms. The following section shows how the two distance/depth-based variograms were fitted. variograms were fitted.

Figure 5. Experimental distance/relative-depth-based variogram for unified data (264 surveying

Figure 5. Experimental distance/relative-depth-based for unified data (264 When surveying samples, shown in Table 1) from multiple seamounts, withvariogram a relative-depth interval of 0–400 m. samples, shown in Table 1) from multiple seamounts, with a relative-depth interval of 0–400 m. interpolating using this variogram via kriging, the information point search was restricted to Δdp ≤ 400 m. WhenThe interpolating using this variogram via kriging, the information point search was restricted dotted curve exhibits effectiveness of the experimental variogram at describing the spatial selfto ∆dp ≤ 400 m. and The dotted curve exhibits effectiveness experimental variogram at describing correlation the continuity of mineral resources of in the terms of cobalt-rich crust thicknesses on the seamounts in such aand depth spatial self-correlation theinterval. continuity of mineral resources in terms of cobalt-rich crust thicknesses on seamounts in such a depth interval. Furthermore, we have to say that distance/relative-depth-based variogram when one relativedepth interval was involved was observed to be equivalent to the conventional variogram in which Furthermore, we have to restricted say thatindistance/relative-depth-based variogram when one the searching of point pairs was a relative-depth interval.

relative-depth interval was involved was observed to be equivalent to the conventional variogram in which4.3. the searching of point pairs was restricted in a relative-depth interval. Variogram Fitting 4.3.1. Distance/Relative-Depth-Based Variogram Fitting 4.3. Variogram Fitting The distance/relative-depth-based variogram could be fitted with the same method used for traditional spherical model functions [23] as follows:

4.3.1. Distance/Relative-Depth-Based Variogram Fitting

The distance/relative-depth-based variogram could be fitted with the same method used for traditional spherical model functions [23] as follows: h i ( Co + (C − Co ) 1.5h/a − 0.5(h/a) 3 , i f h ≤ a γ(h) = Co + (C − Co )Sph(h/a) = (4) C if h ≥ a

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



C  (C  C o )1.5h / a  0.5(h / a )3 , if h  a (h )  C o  (C  C o )Sph(h / a )   o ,if h  a  C

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(4)

Here, h is equivalent to (dis, ∆depth), a is equivalent to range. Thus, the aforementioned equation can be transformed as follows:to (dis , depth ) , a is equivalent to range. Thus, the aforementioned Here, h is equivalent equation can be transformed as follows:

γ(dis, ∆depth) = Co + (C − Co ) × (1.5 × dis/range − 0.5 × (dis/range)3 ) (dis ,depth )  C o  (C  C o )  (1.5  dis / range  0.5  (dis / range )3 )

(5)

(5)

when dis ≤ range and γ(dis, ∆depth) = c when dis > range. where C = 1.02, C0 = 0.4, range = 12,500 m, dis shown range .inwhere  range and ∆depth = 0–400 m are deduced the Figure when dis and when C 4. = 1.02, C0 = 0.4, range = (dis ,from depth )  cvariograms 12,500 m, and Δdepth = 0–400 m are deduced from the variograms shown in Figure 4.

4.3.2. Distance/Gradient-Based Variogram Fitting 4.3.2. Distance/Gradient-Based Variogram Fitting

Gradient/Range Function 

Gradient/Range Function

Only when gradient ranges are fitted to a continuous function can we consider the fitting of our Only when gradient rangeskriging are fittedinterpolation. to a continuousBoth function can we consider the fitting of Deutsch our theoretical variogram and apply Gendzwill and Stauffer and theoretical variogram and apply kriging interpolation. Both Gendzwill and Stauffer and Deutsch and and Journel [23,24] attempted to fit the anisotropy of ranges with spheroidal functions using polar Journel [23,24] attempted to fit the anisotropy of ranges with spheroidal functions using polar coordinates. However, we used the Cartesian coordinate system and three groups of gradients/ranges: coordinates. However, we used the Cartesian coordinate system and three groups of (1.2◦ , 12,500 m), (3.0◦ , 8300 m), and (17.5◦ , 4200 m). We obtained these results using the experimental gradients/ranges: (1.2°, 12,500 m), (3.0°, 8300 m), and (17.5°, 4200 m). We obtained these results using variogram function described Equation (1) and shown in(1)Figure 4, where the range is virtually the experimental variogram by function described by Equation and shown in Figure 4, where the inversely proportional to the gradient. Furthermore, two of the groups were used as control points range is virtually inversely proportional to the gradient. Furthermore, two of the groups were used for fittingastocontrol the following points forequation: fitting to the following equation: range = a + p/( g + 1) (6) range  oa  p /( g  1) o

(6)

where a0 and p are constants, whereas the third group serves as an inspection point after the fitting. 0 and Finally, deduced the gradient/range fitting function where awe p are constants, whereas the third group servesas asfollows: an inspection point after the fitting. Finally, we deduced the gradient/range fitting function as follows:

range = 3080 + 20725/( g + 1) range  3080  20725 /( g  1)

(7) (7)

and the corresponding diagram is shown in Figure 6.

and the corresponding diagram is shown in Figure 6.

Figure 6. Fitted gradient/range function.

Figure 6. Fitted gradient/range function.



Variogram Function Fitting

Variogram Function Fitting The gradient/range function expressed by Equation (6), obtained by fitting, was substituted into the theoretical variogram model represented by Equations (7) and (8), which was proposed by Deutsch

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and Journel [23] as a spherical, an exponential, or a Gaussian model. For our demonstration, we used The gradient/range function expressed by Equation (6), obtained by fitting, was substituted into the spherical model function. the theoretical variogram model represented by Equations (7) and (8), which was proposed by Deutsch and Journel [23] as spherical, exponential, orSph a Gaussian model. γ(adis, g) = Can (C − C0 ) × (dis/range ) For our demonstration, o+ we used the spherical model function.

that is: that is:

(dis ,g )  C o  (C  C 0 )  Sph(dis / range ) γ(dis, g) = Co + (C − Co ) × (1.5 × dis/range − 0.5 × (dis/range)3 )

(8)

(8)

(9)

when dis ≤ range and γ(dis, g) = c when dis > range. where C is the sill value or variance contribution, (dis ,g )  C o  (C  C o )  (1.5  dis / range  0.5  (dis / range )3 ) (9) which in this example is 1.02, and C0 is the nugget effect, set to 0.4. These values were deduced from range in when dis shown and the variograms Figure (dis3. ,g )  c when dis  range . where C is the sill value or variance contribution, which in this of example is 1.02, and C0 is the nugget effect, set to 0.4.isThese were to a The prominent feature the above-mentioned theoretical variogram that values it is subject deduced from the variograms shown in Figure 3. gradient rather than to an orientation; therefore, it is a function of distance and gradient. The prominent feature of the above-mentioned theoretical variogram is that it is subject to a gradient rather than to an orientation; therefore, is a function 4.4. Estimating the Cobalt-Rich Crust Thicknesses onitGuyot Mk of distance and gradient.

With the distance/relative-depth-based variogram, 4.4. Estimating the Cobalt-Rich Crust Thicknesses on Guyot Mk we used ordinary kriging interpolation

method on a 3 × 3 block with 4472 × 4472 m unit squares and a horizontal area of 20 km2 , which is With the distance/relative-depth-based variogram, we used ordinary kriging interpolation equivalent to the minimum grid unit area required by the Regulations on Prospecting and Exploration method on a 3 × 3 block with 4472 × 4472 m unit squares and a horizontal area of 20 km2, which is for Cobalt-Rich Crusts issued by the International Seabed Authority (ISA) [18]. Each location equivalent to the minimum grid unit area required by the Regulations on Prospecting andhad a search radius of 12,000 m. Information points by from were used(ISA) in the interpolation Exploration for Cobalt-Rich Crusts issued theneighboring Internationallocations Seabed Authority [18]. Each (a maximum of nine and a minimum of three). location had a search radius of 12,000 m. Information points from neighboring locations were used in the interpolation maximum of nine and a minimum of three). Natural logarithm(avalues of 77 sampling data points for guyot Mk were taken as information Natural logarithm values of 77of sampling datacrust points for guyot Mk takencells as information points. The natural logarithm values cobalt-rich thicknesses forwere 169 grid were estimated points. The natural logarithm values ofgeostatistical cobalt-rich crust thicknesses for 169 grid cells were estimated via kriging interpolation and using our scheme. Antilog values of the estimated values via kriging interpolation and using our geostatistical scheme. Antilog values of the estimated values are shown in Figure 7 as the distribution of the cobalt-rich crust thicknesses. are shown in Figure 7 as the distribution of the cobalt-rich crust thicknesses.

Figure 7. Distribution of cobalt-rich crust thicknesses estimated using the proposed method. The

Figure 7. Distribution of cobalt-rich crust thicknesses estimated using the proposed method. triangles in blue are the sample locations that are surveyed; almost all the samples locate the The triangles blue are the sample are surveyed; almost all the samplesbylocate bathymetryincontour interval at 1500locations m to 3500that m. This indicates that the values estimated the the bathymetry contour interval at 1500should m to 3500 m. This indicates that the values estimated by the kriging for grid cells in this interval be considered. kriging for grid cells in this interval should be considered.

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5.1. Experimental Variogram on Unification Seamounts Compared with that on a Single Seamount 5.1. Experimental Variogram on Unification Seamounts Compared with that on a SingleofSeamount The same parameters of searching point pairs, i.e., lag separation distance 3000 m, lag tolerance of 1500 m,The relative interval of 0–400 point m, arepairs, usedi.e., in following two distance cases. When datasets same depth parameters of searching lag separation of 3000 m, lag from of 1500 m,Mk relative depth interval 0–400 used followingthe twoexperimental cases. When datasets singletolerance seamount (e.g., guyot, shown in of Table 1)m, isare used toin calculate variogram, from single seamount (e.g., Mk guyot, shown in Table 1) is used to calculate the experimental the number of point pairs in nine lags are 6–74, and the distribution of scatter dots exhibits the variogram,of thethe number of point pairs in nine lags are 6–74, and the scatter dots ineffectiveness experimental variogram (shown in Figure 8).distribution However, of datasets fromexhibits unification the ineffectiveness of the experimental variogram (shown in Figure 8). However, datasets from seamounts (shown in Table 1) show that the number of point pairs in nine lags are 202–497, and the unification seamounts (shown in Table 1) show that the number of point pairs in nine lags are 202–497, dotted curve exhibits the effectiveness of the experimental variogram (shown in Figure 5). and the dotted curve exhibits the effectiveness of the experimental variogram (shown in Figure 5).

Figure 8. Experimental distance/relative-depth-based variogram for data on single seamount (77

Figure 8. Experimental variogram for data on single seamount surveying samples on distance/relative-depth-based guyot Mk, shown in Table 1), with a relative-depth interval of 0–400 m. (77 surveying samples on dots guyot in Table 1), with a relative-depth interval of 0–400 m. Distribution of scatter on Mk, Fig. shown 8 exhibits the ineffectiveness of the experimental variogram Distribution ofthe scatter dots on Figure 8and exhibits the ineffectiveness of the in experimental variogram describing spatial self-correlation the continuity of mineral resources terms of cobalt-rich describing the spatialonself-correlation anda depth the continuity of distribution mineral resources in terms ofatcobalt-rich crust thicknesses seamounts at such interval, the is also noneffective other intervals. crustdepth thicknesses on seamounts at such a depth interval, the distribution is also noneffective at other depth intervals. The ineffectiveness of the experimental variogram (shown in Figure 8) was probably arising because of the insufficient number of surveying data on single seamount. Unifying datasets from The ineffectiveness the experimental variogram (shown in Figure 8) effectiveness was probably arising unification seamounts of brought sufficient number of data, thereby figuring out the of the because of the insufficient number of surveying data on single seamount. Unifying datasets from experimental variogram (shown in Figure 5).

unification seamounts brought sufficient number of data, thereby figuring out the effectiveness of the 5.2. Kriging Compared with OtherinInterpolation experimental variogram (shown Figure 5). Methods on Surveying Locations We compared the method with several interpolation methods. Due to the sparse distribution of

5.2. Kriging Compared with Other Interpolation Methods on Surveying Locations the sampling stations, assigning average values for all survey data (i.e., for guyot Mk, 58 mm in Table 1) to allcompared grid nodes the is a common for evaluating the mineral resources on to seamounts [18].distribution All the We method method with several interpolation methods. Due the sparse data were supposed to be predicted by this average value. As a result, the estimated average error of the sampling stations, assigning average values for all survey data (i.e., for guyot Mk, 58 mm in was equal to the standard deviation of all data (i.e., for guyot Mk, 33 mm in Table 1), and the relative Table 1) to all grid nodes is a common method for evaluating the mineral resources on seamounts [18]. error was 56.9% (i.e., 33 mm/58 mm × 100%). These values served as reference points for evaluating All the data were supposed to be predicted by this average value. As a result, the estimated average the interpolation effects with other more complex methods. error wasInequal standard of all datawith (i.e., for guyot Mk, 33interpolation, mm in Tablewe 1),also and the orderto tothe compare thesedeviation interpolation effects those of our kriging relative error was 56.9% (i.e., 33 mm/58 mm × 100%). These values served as reference points considered the window averaging, the inverse distance weighting, and distance/gradient-based for evaluating the interpolation effects withthe other complexparameters methods. (i.e., the same search radius variogram kriging methods. We used samemore interpolation In order compare interpolation effects with of our kriging interpolation, we also (here, it isto12 km) andthese information point number) with those each method. Each location with known surveying was estimated with the eachinverse of them. distance weighting, and distance/gradient-based considered thedata window averaging, interpolation not encircled by information points so search that they variogramInvalid kriging methods. stations We usedwere the sometimes same interpolation parameters (i.e., the same radius could not form a proper interpolation. Furthermore, in some cases, the locations and their (here, it is 12 km) and information point number) with each method. Each location with known neighboring information points were at different slopes, and they were considered as either impacted surveying data was estimated with each of them. Invalid interpolation stations were sometimes not encircled by information points so that they could not form a proper interpolation. Furthermore, in some cases, the locations and their neighboring

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by a landslide or covered sediments. Herein, the were natural mechanical processes of seamount slope information points were atby different slopes, and they considered as either impacted by a landslide development destroyed the continuity of the cobalt-rich crusts, which resulted to be unpredictable or covered by sediments. Herein, the natural mechanical processes of seamount slope development via interpolation in some locations. Unlike invalid interpolation stations,via each effective destroyed the continuity of the cobalt-rich crusts,the which resulted to be unpredictable interpolation interpolation station was encircled by information points and formed a sound interpolation. in some locations. Unlike the invalid interpolation stations, each effective interpolation station was Furthermore, the information points were alla located on relatively stable slopes without obvious encircled by information points and formed sound interpolation. Furthermore, the information landslide or all sediment history.stable slopes without obvious landslide or sediment cover history. points were locatedcover on relatively Finally, we obtained interpolated effective interpolation interpolation stations, Finally, we obtained interpolated values values from from 28 28 effective stations, which which was was aa subset subset of of the the survey survey stations. stations. For For these these stations, stations, the the data data obtained obtained by by surveying surveying and and forecasting forecasting via via spatial interpolation were used to generate scatter diagrams with the three interpolation methods, as spatial interpolation were used to generate scatter diagrams with the three interpolation methods, shown in Figure 9. as shown in Figure 9.

Figure 9. Cont.

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Figure 9. Cobalt-rich crust thicknesses known and estimated at the 28 survey locations with four Figure 9.interpolation Cobalt-rich crust thicknesses known estimated atpoint the 28 locations witherror four different methods. Estimated error and variance of each is survey annotated by vertical different interpolation methods. Estimated error variance of each point is annotated by vertical error bar. The average error, relative error, and correlation coefficient R between the two sets of values are bar. The average error, relative error, and correlation coefficient R between the two sets of values indicated. are indicated.

The forecasting abilities differed markedly between geostatistical and nongeostatistical methods. Two kriging methods both markedly showed the smallergeostatistical average andand relative errors and the larger The forecasting abilities differed between nongeostatistical methods. correlation coefficient than nongeostatistical methods. Compared with distance/gradient-based Two kriging methods both showed the smaller average and relative errors and the larger correlation variogram kriging, distance/relative-depth-based variogram showed similar abilities on coefficient than nongeostatistical methods. Compared with kriging distance/gradient-based variogram forecasting surveying data. Survey samples are expensive; therefore, incorporating kriging kriging, distance/relative-depth-based variogram kriging showed similar abilities on the forecasting interpolation into our geostatistical scheme may play an important role in evaluating cobalt-rich surveying data. Survey samples are expensive; therefore, incorporating the kriging interpolation into crusts on seamounts. our geostatistical scheme may play an important role in evaluating cobalt-rich crusts on seamounts.

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5.3. Kriging Compared with the Distance/Gradient-Based Variogram Kriging on 159 Grid Cells

5.3. Kriging Compared withinterpolated the Distance/Gradient-Based Variogram Kriging on 159variogram Grid Cells kriging was 169 grid cells were when distance/relative-depth-based used,169 whereas 182 were grid cells were interpolated when distance/gradient-based variogram kriging was grid cells interpolated when distance/relative-depth-based variogram kriging was used, used. From these grid cells, 159 when grid cells exhibit two values that can be kriging estimated the whereas 182 among grid cells were interpolated distance/gradient-based variogram wasby used. two aforementioned kriging methods. By comparing the two values obtained from 159 grid cells From among these grid cells, 159 grid cells exhibit two values that can be estimated by the two in Figure 10, wekriging can infer that theByinterpolation effects of the obtained aforementioned methods were aforementioned methods. comparing the two values from 159kriging grid cells in Figure 10, similar deviation was smaller when distance/relative-depth-based kriging we can even infer though that thethe interpolation effects of the aforementioned kriging methods variogram were similar even was used. though the deviation was smaller when distance/relative-depth-based variogram kriging was used.

Figure 10. The estimated values of cobalt-rich crust thicknesses at 159 grid cells using two different Figure 10. The estimated values of cobalt-rich crust thicknesses at 159 grid cells using two different kriging interpolation methods. The average, deviation, and correlation coefficient, R, between the two kriging interpolation methods. The average, deviation, and correlation coefficient, R, between the two sets of values are indicated. sets of values are indicated.

6. Conclusions 6. Conclusions Here, we presented an alternative tool to the distance/gradient-based variogram (i.e., the Here, we presented an alternative the distance/gradient-based variogramestimated (i.e., the distance/relative-depth-based variogram).tool The to distribution of cobalt-rich crust thicknesses distance/relative-depth-based variogram). The distribution of cobalt-rich crust thicknesses estimated using this alternative variogram was coherent with that estimated using the distance/gradient-based using thisitalternative one, and was more variogram convenientwas andcoherent easier. with that estimated using the distance/gradient-based one, and it was more convenient and easier. The application example of the Magellan seamounts demonstrated that both distance/depthapplication example the Magellan demonstrated both distance/depth-based basedThe variograms can play anofimportant roleseamounts in expressing the spatial that continuity of cobalt-rich crusts variograms can play an important role in expressing the spatial continuity of cobalt-rich on seamounts. The distribution of the survey stations on single seamounts is often too sparse tocrusts allow on seamounts. The distribution of the survey stations on single seamounts is often sparse to variogram fitting; however, data from several adjacent seamounts with similartoo distribution allow variogram however, from several seamounts similar distribution characteristics forfitting; cobalt-rich crustdata thicknesses can adjacent be unified to obtainwith a sufficient dataset for characteristics for cobalt-rich crust thicknesses can be unified to obtain a sufficient dataset for variogram application and fitting. The variogram obtained using such data can be applied variogram to a single application and fitting. The variogram obtained using such data can be applied to a single seamount seamount using the proposed kriging interpolation method. usingAs the proposed kriging method.it was proven that the kriging interpolation method a component of our interpolation geostatistical scheme, As a component of our geostatistical scheme, was proven that the krigingmethods. interpolation can improve the interpolation results compared with it nongeostatistical interpolation Since method can improve the interpolation results compared with nongeostatistical interpolation the survey samples are expensive to obtain and the location data regarding seamounts aremethods. sparse, Since the survey samples areimportant expensiverole to obtain and the location data regarding seamounts are sparse, our approach may play an in evaluating cobalt-rich crusts on seamounts. our approach may play an important role in evaluating cobalt-rich crusts on seamounts. The ranges in different slope gradients could provide reference data for the design of sampling sites when surveying seamounts. Our results suggest that the spacing of these sampling sites should be smaller than the given ranges (i.e., 6000–12,000 m for bathymetric contour orientations and 2000–4000 m for gradient orientations). Our method could be effectively used not only for guyots but also for steeple

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The ranges in different slope gradients could provide reference data for the design of sampling sites when surveying seamounts. Our results suggest that the spacing of these sampling sites should be smaller than the given ranges (i.e., 6000–12,000 m for bathymetric contour orientations and 2000–4000 m for gradient orientations). Our method could be effectively used not only for guyots but also for steeple top seamounts and mountain slopes on land. Moreover, it could be used to determine not only cobalt-rich crust thicknesses but also other parameter values, such as crustal element concentrations. We plan to study these aspects in the future. The spatial continuity of cobalt-rich crusts on seamount slopes was found to form the basis for spatial interpolation. At some locations, this continuity may be destroyed by natural mechanical processes. Therefore, when interpolating for mineral resource evaluation using the kriging interpolation or similar methods, it is important to take into account a variety of factors (e.g., the slope stability and the sediment cover history). Author Contributions: D.D. conceived the paper, and designed the arithmetic; S.Y. performed data analyzation, and edited the document; F.Y. edited the document; Z.Z. performed data analyzation; Q.S. drew figures; G.Y. collected data, and performed data analyzation. All authors prepared the manuscript together. Funding: This research is funded by the National Natural Science Foundation of China (no. 41776076), Ocean Mineral Resources R&D Association (DY135-S2-2-02, C1-1-01 and DY125-13-R-03). Conflicts of Interest: There are no conflicts of interests.

References 1.

2.

3.

4. 5. 6. 7. 8. 9. 10. 11. 12. 13. 14.

Hein, J.R. Cobalt-rich ferromanganese crusts: Global distribution, composition, origin and research activities. In Minerals Other than Polymetallic Nodules of the International Seabed Area, Proceedings of the International Seabed Authority Workshop, Kingston, Jamaica, 26–30 June 2000; International Seabed Authority: Kingston, Jamaica, 2002; Volume 2, pp. 36–89. Available online: https://www.researchgate.net/publication/264382918 (accessed on 29 August 2018). Hein, J.R.; Koschinsky, A.; Bau, M.; Manheim, F.T.; Kang, J.K. Cobalt-rich ferromanganese crusts in the Pacific. In Handbook of Marine Mineral Deposits, 1st ed.; Cronan, D.S., Ed.; CRC Press: Boca Raton, FL, USA, 1999; Volume 2, pp. 239–279. Available online: https://www.researchgate.net/publication/264383136 (accessed on 29 August 2018). Verlaan, P.A.; Cronan, D.S.; Morgan, C.L. A comparative analysis of compositional variations in and between marine ferromanganese nodules and crusts in the South Pacific and their environmental controls. Prog. Oceanograp. 2004, 63, 125–158. [CrossRef] Carr, J.R. Geostatistics for natural resources evaluation. Environ. Eng. Geosci. 1998, 4, 278–279. [CrossRef] Matheron, G. La Theorie Des Variables Regionaliseesetses Applications; Ecole Nat. Sup. Des Mines: Paris, France, 1971. Journel, A.G.; Huijbregts, C.J. Mining Geostatistics; Academic Press: London, UK, 1978. Usui, A.; Someya, M. Distribution and composition of marine hydrogenetic and hydrothermal manganese deposits in the Northwest Pacific. Geol. Soc. Lond. Spec. Publ. 1997, 119, 177–198. [CrossRef] Zhang, F.Y.; Zhang, W.; Zhu, K.; Zhang, X.; Zhu, B. Distribution characteristics of cobalt-rich ferromanganese crust resources on submarine seamounts in the Western Pacific. Acta Geol. Sin. 2008, 82, 796–803. Armstrong, M. Improving the Estimation and Modelling of the Variogram. In Geostatistics for Natural Resources Characterization; Springer: Dordrecht, The Netherlands, 1984. Cressie, N.A. Statistics for Spatial Data; Wiley: New York, NY, USA, 1993. Cressie, N.; Hawkins, D.M. Robust estimation of the variogram: I. J. Int. Assoc. Math. Geol. 1980, 12, 115–125. [CrossRef] Isaaks, E.H.; Srivastava, R.M. Spatial continuity measures for probabilistic and deterministic geostatistics. Math. Geol. 1988, 20, 313–341. [CrossRef] Isaaks, E.H.; Srivastava, R.M. An Introduction to Applied Geostatistics; Oxford University Press: New York, NY, USA, 1989; pp. 483–485. Journel, A.G. Geostatistics for the Environmental Sciences: An Introduction; US Environmental Protection Agency, Environmental Monitoring Systems Laboratory: Washington, DC, USA, 1987.

Minerals 2018, 8, 374

15. 16. 17. 18.

19. 20.

21.

22. 23. 24.

15 of 15

Journel, A.G. New distance measures: The route toward truly nongaussian geostatistics. Math. Geol. 1988, 20, 459–475. [CrossRef] Omre, H. The variogram and its estimation. In Geostatistics for Natural Resources Characterization; Springer: Dordrecht, The Netherlands, 1984; pp. 107–125. [CrossRef] Du, D.; Wang, C.; Du, X.; Yan, S.; Ren, X.; Shi, X.; Hein, J.R. Distance-gradient-based variogram and kriging to evaluate cobalt-rich crust deposits on seamounts. Ore Geol. Rev. 2017, 84, 218–227. [CrossRef] Hein, J.R.; Conrad, T.A.; Dunham, R.E. Seamount characteristics and mine-site model applied to exploration and mining-lease-block selection for cobalt-rich ferromanganese crusts. Mar. Geores. Geotechnol. 2009, 27, 160–176. [CrossRef] Du, D.; Ren, X.; Yan, S.; Shi, X.; Liu, Y.; He, G. An integrated method for the quantitative evaluation of mineral resources of cobalt-rich crusts on seamounts. Ore Geol. Rev. 2017, 84, 174–184. [CrossRef] Gousie, M.B.; Franklin, W.R. Converting Elevation Contours to a Grid; Dept of Geography Simon Fraser University: Burnaby, BC, Canada, 1998; pp. 647–656. Available online: https: //www.researchgate.net/profile/Michael_Gousie/publication/2318287_Converting_Elevation_Co ntours_to_a_Grid/links/0deec52cb05f8df45f000000.pdf (accessed on 29 August 2018). Jean-Laurent, M. Book Review: Geomodeling, Mathematical Geology; Springer: Berlin, Germany, 2004; Volume 36, pp. 283–286. Available online: https://link.springer.com/article/10.1023%2FB%3AMATG.0000020673.98 166.ea (accessed on 29 August 2018). Deutsch, C.V.; Wang, L. Quantifying object-based stochastic modeling of fluvial reservoir. Math. Geol. 1996, 28, 857–880. [CrossRef] Deutsch, C.V.; Journel, A.G. GSLIB Geostatistical Software Library and User’s Guide; Oxford University Press: New York, NY, USA, 1998. Gendzwill, D.J.; Stauffer, M.R. Analysis of triaxial ellipsoids: Their shapes, plane sections, and plane projections. J. Int. Assoc. Math. Geol. 1981, 13, 135–152. [CrossRef] © 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).