# Empirical Application of Generalized Rayleigh Distribution for Mineral Resource Estimation of Seabed Polymetallic Nodules

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Three Hypotheses for an Idealized Model of Seabed Polymetallic Nodules

- Each nodules piece is of ellipsoidal shape (e.g., in Figure 2a,b), which is defined by its three axes ${X}_{i}$, ${Y}_{i}$ and ${Z}_{i}$, where $i=1,\text{}2,\text{}3\dots N$, with $N$ being the number of nodules. Here ${X}_{i}$ is the long or major axis, which is usually in the horizontal plane while ${Y}_{i}$ and ${Z}_{i}$ are the two typically shorter minor axes in the horizontal and vertical planes.
- Within a certain boundary (domain) on the seabed, the ellipsoidal nodules are similar in shape, i.e., the ratio between two minor axes and the major axis, ${\epsilon}_{1}=\frac{{Y}_{i}}{{X}_{i}}$ and ${\epsilon}_{2}=\frac{{Z}_{i}}{{X}_{i}}$ are constant.
- Within a certain boundary (domain) on the seabed, the long axis of nodule ${X}_{i}$ follows a Generalized Rayleigh Distribution (GRD), which is defined by a pair of parameters α and β (See Section 3).

## 3. Generalized Rayleigh Distribution (GRD) and the Traditional Method

#### 3.1. Mean and Standard Deviation of the Generalized Rayleigh Distribution (GRD)

#### 3.2. Test of Goodness-of-Fit of Generalized Rayleigh Distribution

#### 3.2.1. Parameter Estimation by Maximum Likelihood Estimation (MLE)

#### 3.2.2. The Anderson–Darling Test Statistics

#### 3.2.3. The Test Criteria for Hypothesis

## 4. A New Empirical Method and Its Application to Nodule Resources

#### 4.1. Empirical Estimation of Parameters of the Generalized Rayleigh Distribution

#### 4.2. Resource Estimation for Seabed Polymetallic Nodules Using Coverage and Abundance

- The nodule is assumed to be in an idealized ellipsoid shape.
- Nodules within a certain boundary are assumed to be “similar” in shape, the ratios between the lengths of the two minor axes and the major axis (denoted by ${\epsilon}_{1}$ and ${\epsilon}_{2}$) are constant.

#### 4.2.1. Prediction of Nodule Coverage: Idealized Nodules

#### 4.2.2. Prediction Nodule Abundance I: Based on “Idealized Nodule” Model

#### 4.2.3. Prediction Nodule Abundance II: With Empirical Long-Axis-Weight Relationship

#### 4.2.4. Relation between Nodule Percentage Coverage and Abundance

- Equations (15) and (19) can be used independently to calculate the nodule percentage coverage ${C}_{N}$ and the abundance ${A}_{N}$, respectively.
- If an estimation of the ${C}_{N}$ is already estimated (e.g., using digitization technique from seabed imagery), then Equation (28) or Equation (29) can be used to compute ${A}_{N}$.

## 5. Test of Hypotheses of the Idealized Nodule Model

#### 5.1. Test of Hypothese 1 and 2: Linear Regression Analyses on Nodule Dimensions and Weights

- Case 1: Nodule long axis and its horizontal minor axis;
- Case 2: Nodule long axis and its vertical minor axis;
- Case 3: Nodule weight and its volume; and
- Case 4: Nodule long axis and its weight.

^{2}> 95%, a great majority (>95%) of the data points supports the hypothesis that, with a certain boundary, the ratios between the length of the major axes X and the lengths of the horizontal and the vertical minor axes Y and Z can be considered as constants. It is also noted the ratios vary between regions. The result from the 3rd Case indicates a nodule density of 1.93 g/cm

^{3}although it is only supported by a small sample. The Case 4 indicates the nodule weight is strongly correlated with the 2.5056th and 2.7210th power of the long axes, supported by about 88% and 93% of the data points from the two regions, respectively. It is worthwhile to notice, for “idealized nodules”, the nodule weight is proportional exactly to the cubic (the 3rd power) of its long axis.

#### 5.2. Test of Hypothses 3: Goodness-of-Fit Test of Generalized Rayleigh Distribution for Nodule Long Axes

#### 5.3. Comments on the Level of Support

## 6. Numerical Results of Nodule Resource Prediction Using the New Empirical Method

#### 6.1. Sample Datasets

- 1.
- Dataset 1: regional scale box-core sample dataset (physical weights). This involves four TOML exploration contract areas (TOML B, C, D, F; Figure 9) spanning some 2000 km of longitude and 700 km of latitude. The dataset thus allows for examination of a general relationship.
**2.**- Dataset 2: local scale box-core sample dataset (physical weight) of two distinct facies types but only within the TOML F area (~200 × 200 km). Type 1 nodules are smaller and often densely packed, type 2 nodules are significantly larger and more variable (cf. [5]). The dataset thus allows for differences in nodule types from an area where the distinction between type is simple and straightforward.
**3.**- Dataset 3: two local scale towed photo sample datasets (long-axis abundance estimate) between the TOML B and C areas (~300 km apart). The dataset is limited in that actual nodule weights cannot be compared, but it allows for larger datasets from two distinctly different areas to be compared.

#### 6.2. Prediction of Abundance of Seabed Nodules

- Chart (a) showing ratios based on abundance calculated directly by the empirical formula Equation (19), which is strictly based on the three hypotheses for idealized nodule model in Section 2. The axis ratios ${\epsilon}_{1}$ and ${\epsilon}_{2}$ used in the formula are extracted from the analyses of BGR East and GSR Central data (Table 1 in Section 5.1);
- Chart (b) showing the ratios in Chart (a) corrected by a “linear adjustment”. Each individual ratio in Chart (a) is factored/divided by the result of linear regression of the ratios, and the corrected results are shown in Chart (b); and
- Chart (c) showing ratios based on abundance calculated by the empirical formula Equation (19), incorporating the long-axis-weight relationship observed by several researchers (e.g., Felix [13]), which indicates the nodule weight is coorelated to the 2.7–2.8
^{th}power of its long-axis (noting for “idealized nodule”, it is the 3rd power).

## 7. Conclusions

- There is statistically significant evidence that the forms of CCZ polymetallic nodules resemble an “idealized nodule” model based on three hypotheses: (1) broadly ellipsoidal shape, (2) similar forms between nodules in a given area and (3) the nodule long axes follow a two-parameter Generalized Rayleigh Distribution (GRD). These three hypotheses were tested using field measurements from available nodule samples collected from CCZ. Numerical evidence supports the three hypotheses, possibly due to the relatively stable seabed environment and the long growth period of the nodules removing short-term transient effects.
- The distribution of nodules sizes and associated parameters can be estimated using empirical formulae. Specifically, explicit empirical formulae have been derived for direct calculation of GRD parameter α and β (Equation (11)), for percentage coverage C
_{N}(Equation (15)), and for abundance A_{N}(Equation (19) or Equation (25)). These formulas are found to be sufficiently accurate for mineral resource estimation and are much easier to use than the traditional analytical methods for GRD. - The direct application of the formula for A
_{N}does display a slight bias of over-estimating the abundance for larger nodules. However, unbiased accurate prediction of nodule abundance can be achieved by applying either a “linear adjustment” or a long-axis-weight relationship. - For two of the TOML areas the new empirical method provides close agreement but from the third area there is a consistent offset. This may be related to the degree of clay-ooze sediment cover in that third area. Analyses of samples from other regions will be needed to better understand the generality of the empirical model and its derived formulae. Such analysis is needed in any event to calibrate the model in other areas.
- The new empirical method with derived explicit formulae has shown the potential of achieving more accurate mineral resource estimation with reduced sample numbers and sizes. The new understanding of the nodule size distribution can likely also improve the efficiency of design and configuration of mining equipment with limitations regarding particle size.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Functions ${\mathit{F}}_{\mathbf{1}}\mathbf{\left(}\mathit{\alpha}\mathbf{\right)}$, ${\mathit{F}}_{\mathbf{2}}\mathbf{\left(}\mathit{\alpha}\mathbf{\right)}$ and ${\mathit{F}}_{\mathbf{3}}\mathbf{\left(}\mathit{\alpha}\mathbf{\right)}$

## Appendix B. Computation of Critical Values for Goodness-of-Fit Test of Generalized Rayleigh Distribution by Monte Carlo Simulations

- A set of random numbers of size n is generated in the interval $\left(0,1\right)$ as values of cumulative distribution function (CDF). Equation (2) is used to back-calculate a sample ${X}_{1},{X}_{2},\dots ,{X}_{n}$of size n for given $\alpha $ and $\beta $.
- For a sample ${X}_{1},{X}_{2},\dots ,{X}_{n}$, Equations (5) and (6), based on MLE method, are solved iteratively to estimate parameters $\widehat{\alpha}$ and $\widehat{\beta}$.
- Parameters $\widehat{\alpha}$ and $\widehat{\beta}$ are used in Equation (2) to calculate ${z}_{i}=F\left({x}_{i}\right)$, with values in ascending order.
- Equation (7) is used to calculate test statistics ${A}_{n}^{2}$and ${V}_{n}^{2}$, using values of ${z}_{i}$ calculated in step 3.
- Steps 1. to 4. above are repeated to generate a sample for ${A}_{n}^{2}$and ${V}_{n}^{2}$.
- The percentiles of ${A}_{n}^{2}$and ${V}_{n}^{2}$ are calculated as critical values. The $\left(1-\gamma \right)$
^{th}percentile is taken as the critical value for the level of significance of γ.

**Table A1.**Critical values$\text{}{y}_{\gamma}$ and ${u}_{\gamma}$ for various shape parameters $\mathit{\alpha}$ and sample sizes $\mathit{n}$.

Shape Parameter α | 1.0 | 1.5 | 2.0 | 2.5 | 3.0 | 3.5 | 4.0 | ||||||||||||||||
---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|---|

Significance Level γ | 10% | 5% | 1% | 10% | 5% | 1% | 10% | 5% | 1% | 10% | 5% | 1% | 10% | 5% | 1% | 10% | 5% | 1% | 10% | 5% | 1% | ||

Sample Size n | 100 | ${y}_{\gamma}$ | 0.66 | 0.79 | 1.10 | 0.65 | 0.78 | 1.08 | 0.64 | 0.77 | 1.07 | 0.64 | 0.76 | 1.06 | 0.64 | 0.76 | 1.05 | 0.64 | 0.76 | 1.05 | 0.64 | 0.76 | 1.05 |

${u}_{\gamma}$ | 0.34 | 0.41 | 0.58 | 0.34 | 0.41 | 0.57 | 0.33 | 0.41 | 0.57 | 0.33 | 0.40 | 0.57 | 0.33 | 0.40 | 0.57 | 0.33 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | ||

200 | ${y}_{\gamma}$ | 0.66 | 0.79 | 1.11 | 0.65 | 0.78 | 1.08 | 0.65 | 0.77 | 1.07 | 0.64 | 0.77 | 1.05 | 0.64 | 0.76 | 1.05 | 0.64 | 0.76 | 1.06 | 0.64 | 0.76 | 1.05 | |

${u}_{\gamma}$ | 0.34 | 0.41 | 0.58 | 0.34 | 0.41 | 0.58 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | 0.34 | 0.40 | 0.57 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | ||

300 | ${y}_{\gamma}$ | 0.66 | 0.79 | 1.10 | 0.65 | 0.78 | 1.08 | 0.65 | 0.77 | 1.08 | 0.64 | 0.77 | 1.06 | 0.64 | 0.76 | 1.05 | 0.64 | 0.76 | 1.05 | 0.64 | 0.76 | 1.06 | |

${u}_{\gamma}$ | 0.34 | 0.41 | 0.58 | 0.34 | 0.41 | 0.58 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | ||

400 | ${y}_{\gamma}$ | 0.66 | 0.80 | 1.11 | 0.65 | 0.78 | 1.08 | 0.65 | 0.77 | 1.07 | 0.64 | 0.77 | 1.06 | 0.64 | 0.76 | 1.06 | 0.64 | 0.76 | 1.06 | 0.64 | 0.76 | 1.05 | |

${u}_{\gamma}$ | 0.34 | 0.41 | 0.59 | 0.34 | 0.41 | 0.58 | 0.34 | 0.41 | 0.58 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | ||

500 | ${y}_{\gamma}$ | 0.66 | 0.80 | 1.11 | 0.65 | 0.78 | 1.09 | 0.64 | 0.77 | 1.07 | 0.64 | 0.77 | 1.06 | 0.64 | 0.77 | 1.06 | 0.64 | 0.76 | 1.06 | 0.64 | 0.76 | 1.05 | |

${u}_{\gamma}$ | 0.34 | 0.41 | 0.59 | 0.34 | 0.41 | 0.58 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 | 0.34 | 0.41 | 0.57 |

**Figure A1.**Critical values$\text{}{y}_{\gamma}$ versus shape parameters $\alpha $ and sample sizes $\mathit{n}$ with 200,000 and 250,000 simulations.

**Figure A2.**Critical values ${u}_{\gamma}$ versus shape parameters $\alpha $ and sample sizes $\mathit{n}$ with 200,000 and 250,000 simulations.

## Appendix C. Determination of Minimal Sample Size for Estimation of Statistical Distribution Using Monte Carlo Simulations

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**Figure 1.**Example towed seabed photo (

**a**) and box-core sample (

**b**). Mounds of clay-ooze without nodules in the seabed photo are caused by bioturbation.

**Figure 4.**${\mathrm{F}}_{1}\left(\alpha \right)$, ${\mathrm{F}}_{2}\left(\alpha \right)$, ${\mathrm{F}}_{3}\left(\alpha \right)\text{}$and $\mathrm{G}\left(\alpha \right)$ as functions of $\alpha :\text{}$Numerical Integration vs. Empirical Formulas.

**Figure 6.**Linear Regression between Nodule Dimensions and Weights for GSR Central Region. Data source [14]. Note in (

**d**) the legend shows the slopes and intercepts of the linear regressions of the data of Felix [13] and TOML [4], where the intercepts are converted from cm to mm (in logarithmic scale).

**Figure 10.**Comparisons of the measured Abundances and the predicted ones by the new empirical method with Dataset 1: (

**a**) by empirical formula Equation (19); (

**b**) by empirical formula Equation (19) with a linear adjustment; (

**c**) by empirical formula Equation (25).

**Figure 11.**Comparisons of the measured Abundances and the predicted ones by the new empirical method with Dataset 2: (

**a**) by empirical formula Equation (19); (

**b**) by empirical formula Equation (19) with a linear adjustment; (

**c**) by empirical formula Equation (25).

**Figure 12.**Comparisons of the measured Abundances and the predicted ones by the new empirical method with Dataset 3: (

**a**) by empirical formula Equation (19); (

**b**) by empirical formula Equation (19) with a linear adjustment; (

**c**) by empirical formula Equation (25).

**Figure 13.**Comparisons of the measured Percentage Coverage and the predicted ones by the new empirical method for Datasets 2 in (

**a**) and Dataset 3 in (

**b**) with predictions by empirical formula Equation (15).

**Figure 14.**Seabed photos from TOML B, C and F. (

**a**) TOML B, CCZ15-F02: 2015_08_17_032745 (

**b**) F05: 2015_08_29_071000 (

**c**) TOML F type 1 nodules CCZ15-B105 (

**d**) TOML F type 2 nodules CCZ15- B99. Images in (

**a**) and (

**b**) are 2.4 m × 1.6 m in area. Trigger weight in images (

**c**) and (

**d**) is 28 cm × 16 cm long.

Case | Regression Parameters | Regions | Sample Size | Estimated Slope | Estimated Intercept | Coefficient of Determination (R ^{2}) | |
---|---|---|---|---|---|---|---|

1 | Minor Axis Y (Horizontal, mm) | Long Axis X (mm) | BGR East | 1376 | 0.7350 | 0 (Forced) | 97.52% |

GSR Central | 259 | 0.7618 | 97.63% | ||||

2 | Minor Axis Z (Vertical, mm) | Long Axis X (mm) | BGR East | 1376 | 0.4762 | 0 (Forced) | 95.09% |

GSR Central | 259 | 0.5389 | 96.83% | ||||

3 | Weight (g) | Volume (cm^{3}) | BGR East | 99 | 1.9269 | 0 (Forced) | 90.93% |

GSR Central | No Data | / | / | / | |||

4 | Weight (Logarithmic, g) | Long Axis X (Logarithmic, mm) | BGR East | 1376 | 2.5067 | −2.6245 | 87.68% |

GSR Central | 259 | 2.7210 | −2.9439 | 93.13% |

No | Sample ID | TOML Area | Type | Sample Size | Mean | Standard Deviation |
---|---|---|---|---|---|---|

1 | 2015_08_10_172643 | B | Towed Photo | 336 | 3.091 | 2.512 |

2 | 2015_08_10_220159 | B | Towed Photo | 153 | 7.767 | 2.387 |

3 | 2015_08_11_121357 | B | Towed Photo | 403 | 5.978 | 1.732 |

4 | 2015_08_29_131349 | C | Towed Photo | 440 | 5.425 | 1.995 |

5 | 2015_09_02_185307 | C | Towed Photo | 113 | 3.827 | 1.270 |

6 | CCZ15-B51 | D | Washed Sample | 67 | 7.486 | 2.404 |

7 | CCZ15-B102 | F | Washed Sample | 278 | 4.318 | 1.705 |

8 | CCZ15-B106 | F | Washed Sample | 559 | 3.681 | 1.298 |

9 | CCZ15-B110 | F | Washed Sample | 135 | 6.910 | 2.602 |

No | Sample ID | α | β | ${\mathit{A}}_{\mathit{n}}^{2}$ | ${\mathit{y}}_{\mathit{\gamma}}$ | ${\mathit{V}}_{\mathit{n}}^{2}$ | ${\mathit{u}}_{\mathit{\gamma}}$ | Conclusions |
---|---|---|---|---|---|---|---|---|

1 | 2015_08_10_172643 | 0.623 | 0.211 | 9.957 | 0.833 | 5.436 | 0.43 | Not Generalized Rayleigh |

2 | 2015_08_10_220159 | 2.714 | 0.162 | 0.66 | 0.784 | 0.257 | 0.414 | Generalized Rayleigh Dist. at 5% Level of Significance |

3 | 2015_08_11_121357 | 3.598 | 0.226 | 0.69 | 0.770 | 0.226 | 0.410 | |

4 | 2015_08_29_131349 | 1.965 | 0.210 | 0.541 | 0.777 | 0.305 | 0.408 | |

5 | 2015_09_02_185307 | 2.690 | 0.327 | 0.159 | 0.789 | 0.076 | 0.419 | |

6 | CCZ15-B51 | 1.701 | 0.144 | 0.376 | 0.804 | 0.200 | 0.423 | |

7 | CCZ15-B102 | 1.396 | 0.243 | 1.435 | 0.790 | 0.605 | 0.412 | Not Generalized Rayleigh |

8 | CCZ15-B106 | 2.410 | 0.321 | 0.738 | 0.778 | 0.314 | 0.410 | Generalized Rayleigh Dist. at 5% Level of Significance |

9 | CCZ15-B110 | 1.890 | 0.171 | 0.361 | 0.791 | 0.192 | 0.418 |

Data-Set | TOML Areas | Number of Samples | Comparative Data Type | Range of Measured Abundances | Range of Mean Long Axes * | Range of Coefficient of Variation * |
---|---|---|---|---|---|---|

1 | B, C, D, F | 2, 3, 7, 3 | Washed sample weights | 3.2 to 25.7 kg/m^{2} | 2.2 to 7.6 cm | 0.23 to 0.86 |

2 | F | 11 for Type 1 9 for Type 2 | Washed sample weights | 1.2 to 21.3 3.3 to 29.1 | 2.2 to 3.9 2.6 to 9.2 | 0.28 to 0.45 0.28 to 0.72 |

3 | B C | 68 85 | Long axis estimates on individual nodule images | 0.03 to 31 0.01 to 18 | 1.6 to 7.8 1.5 to 6.1 | 0.24 to 0.96 0.25 to 0.83 |

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**MDPI and ACS Style**

Yu, G.; Parianos, J.
Empirical Application of Generalized Rayleigh Distribution for Mineral Resource Estimation of Seabed Polymetallic Nodules. *Minerals* **2021**, *11*, 449.
https://doi.org/10.3390/min11050449

**AMA Style**

Yu G, Parianos J.
Empirical Application of Generalized Rayleigh Distribution for Mineral Resource Estimation of Seabed Polymetallic Nodules. *Minerals*. 2021; 11(5):449.
https://doi.org/10.3390/min11050449

**Chicago/Turabian Style**

Yu, Gordon, and John Parianos.
2021. "Empirical Application of Generalized Rayleigh Distribution for Mineral Resource Estimation of Seabed Polymetallic Nodules" *Minerals* 11, no. 5: 449.
https://doi.org/10.3390/min11050449