9.2 Marine Minerals

Step 1: Define Nodule Abundance as a Spatial Field

Nodule abundance $\rho_A(x,y)$ (kg/m$^2$) is a spatially varying quantity measured at discrete box-core sampling stations $(x_i, y_i)$. The total mass of nodules in the exploration area $A$ is:

$$M_{\text{total}} = \int\!\!\int_A \rho_A(x,y) \, dA$$

Step 2: Discretise Using Geostatistical Interpolation

With $n$ sample stations, we estimate $\rho_A$ at unsampled locations using kriging. The kriged estimate is a weighted linear combination of observations:

$$\hat{\rho}_A(\mathbf{x}_0) = \sum_{i=1}^{n} w_i \, \rho_A(\mathbf{x}_i), \quad \sum_i w_i = 1$$

Step 3: Incorporate Metal Grade

Each metal $m$ has a grade (weight fraction) $g_m(x,y)$ that also varies spatially. The total metal resource is the product of abundance and grade integrated over area:

$$M_m = \int\!\!\int_A \rho_A(x,y) \cdot g_m(x,y) \, dA \approx \sum_{k=1}^{N_{\text{cells}}} \hat{\rho}_A(\mathbf{x}_k) \cdot \hat{g}_m(\mathbf{x}_k) \cdot \Delta A_k$$

Step 4: Uncertainty via Kriging Variance

Kriging provides not just the best estimate but also the estimation variance $\sigma_k^2$ at each cell, allowing confidence intervals on the total resource:

$$\text{Var}(M_m) = \sum_{k=1}^{N_{\text{cells}}} \left[\hat{g}_m^2 \sigma_{\rho,k}^2 + \hat{\rho}_A^2 \sigma_{g,k}^2\right] (\Delta A_k)^2$$

Step 1: Radioactive Decay Law

A radioisotope incorporated into the nodule during growth decays exponentially. For $^{10}\text{Be}$ ($t_{1/2} = 1.387$ Myr):

$$N(t) = N_0 \, e^{-\lambda t}, \quad \lambda = \frac{\ln 2}{t_{1/2}}$$

Step 2: Relate Depth in Nodule to Age

If the nodule grows radially at a constant rate $G$ (mm/Myr), then a layer at radial depth $r$ from the surface was deposited at time $t = r/G$ ago. The isotope activity at depth $r$ is:

$$A(r) = A_0 \, \exp\!\left(-\frac{\lambda \, r}{G}\right)$$

Step 3: Linearise by Taking the Logarithm

Taking the natural log gives a linear relationship between $\ln A$ and radial depth $r$. The slope of this line yields the growth rate:

$$\ln A(r) = \ln A_0 - \frac{\lambda}{G} \, r \quad \Longrightarrow \quad G = -\frac{\lambda}{\text{slope}}$$

Step 4: Typical Results

For CCZ nodules, measured $^{10}\text{Be}$ profiles yield slopes corresponding to $G \approx 1\text{--}10$ mm/Myr. Hydrogenous layers (seawater precipitation) grow at ~1--5 mm/Myr, while diagenetic layers (pore water) grow faster at ~5--10 mm/Myr. A 5 cm radius nodule therefore has an age of order 5--50 Myr.