Geostatistics

Geostatistics originated in mining and petroleum exploration (Matheron, 1963) and has since expanded to hydrology, soil science, ecology, and atmospheric science. The central object is the regionalized variable—a random function $Z(\mathbf{s})$ whose realizations are spatially correlated but deterministic at any fixed location.

Regionalized Variables

Semivariogram Theory

Properties of Valid Semivariograms

A function $\gamma(\mathbf{h})$ is a valid semivariogram if and only if it satisfies:

1. $\gamma(\mathbf{0}) = 0$
2. $\gamma(-\mathbf{h}) = \gamma(\mathbf{h})$ (symmetry)
3. $-\gamma(\mathbf{h})$ is conditionally negative definite: for any set of locations and weights $\{w_i\}$ with $\sum_i w_i = 0$, $\sum_i \sum_j w_i w_j \gamma(\mathbf{s}_i - \mathbf{s}_j) \leq 0$.

Parameter Estimation

Common weighting schemes: $w_k = 1$ (equal weights), $w_k = N(\mathbf{h}_k)$ (proportional to pair count), or $w_k = 1/\text{Var}[\hat{\gamma}(\mathbf{h}_k)]$ (inverse variance). Maximum likelihood estimation is asymptotically more efficient but requires distributional assumptions.

Kriging Variants

Simple Kriging (SK)

Assumes known mean $\mu$. The kriging variance $\sigma^2_{SK}(\mathbf{s}_0) = C(0) - \mathbf{c}^\top \mathbf{C}^{-1}\mathbf{c}$ is independent of observed values—it depends only on geometry.

Ordinary Kriging (OK)

The unbiasedness constraint $\sum_i \lambda_i = 1$ eliminates the need for a known mean. The Lagrange multiplier $\psi$ equals $\sigma^2_{OK} - \sigma^2_{SK}$ when the true mean equals $\mu$.

Indicator Kriging

For categorical or non-Gaussian data, indicator kriging works with transformed values:

Each threshold $z_k$ yields a separate kriging system, producing the conditional cumulative distribution function (CCDF) at each unsampled location.

Block Kriging

Block kriging estimates the average value over a support $V$ (a block or panel) rather than at a point:

The block kriging variance satisfies $\sigma^2_{OK}(V) \leq \sigma^2_{OK}(\mathbf{s}_0)$ for any point $\mathbf{s}_0 \in V$—the support effect reduces estimation variance. The block covariance is:

Cross-Validation and Model Validation

Under a correctly specified model:

• Standardized errors $e_i / s_i$ should be approximately $N(0, 1)$
• The mean error $\bar{e} \approx 0$ (unbiasedness)
• The mean squared standardized error $\frac{1}{n}\sum_i (e_i/s_i)^2 \approx 1$
• No correlation between $e_i$ and spatial location

Non-Stationary Geostatistics

Alternatively, the external drift kriging model incorporates auxiliary fields:

where the drift is specified by external variables $f_k(\mathbf{s})$ (e.g., elevation, temperature) observed everywhere.

Resource Estimation

The discrete Gaussian model (DGM) provides the conditional distribution of block grades for resource classification:

This standardized variable is used for local uncertainty propagation via Turning Bands simulation.

Python Implementation

import numpy as np
import pandas as pd
from scipy.optimize import minimize
from scipy.spatial.distance import pdist, squareform, cdist
import matplotlib.pyplot as plt

np.random.seed(42)

# --- Simulate regionalized data ---
n = 150
coords = np.column_stack([
    np.random.uniform(0, 100, n),
    np.random.uniform(0, 100, n)
])

def matern_cov(h, sill, range_param, nu):
    from scipy.special import kv, gamma as gamma_func
    h = np.maximum(h, 1e-10)
    coeff = (2 ** (1 - nu)) / gamma_func(nu)
    arg = np.sqrt(2 * nu) * h / range_param
    return sill * coeff * (arg ** nu) * kv(nu, arg)

# Generate from Matérn covariance
D = squareform(pdist(coords))
C = matern_cov(D, sill=3.0, range_param=20.0, nu=1.5)
C += 1e-6 * np.eye(n)  # nugget for stability
L = np.linalg.cholesky(C)
z = L @ np.random.standard_normal(n)

# --- Empirical variogram ---
def compute_variogram(coords, z, n_bins=20, max_dist=50.0):
    D = squareform(pdist(coords))
    pairs = np.triu_indices(len(z), k=1)
    dists, diffs_sq = D[pairs], (z[pairs[0]] - z[pairs[1]]) ** 2

    bins = np.linspace(0, max_dist, n_bins + 1)
    centers = 0.5 * (bins[:-1] + bins[1:])
    gamma, counts = np.zeros(n_bins), np.zeros(n_bins)

    for i in range(n_bins):
        mask = (dists >= bins[i]) & (dists < bins[i + 1])
        counts[i] = mask.sum()
        if counts[i] > 0:
            gamma[i] = 0.5 * diffs_sq[mask].mean()
        else:
            gamma[i] = np.nan
    return centers, gamma, counts

h_vals, gamma_emp, counts = compute_variogram(coords, z)
valid = ~np.isnan(gamma_emp)

# --- Fit Matérn variogram via weighted least squares ---
def matern_variogram(h, nugget, sill, range_param, nu):
    from scipy.special import kv, gamma as gamma_func
    h = np.maximum(h, 1e-10)
    coeff = (2 ** (1 - nu)) / gamma_func(nu)
    arg = np.sqrt(2 * nu) * h / range_param
    full_sill = nugget + sill
    cov = sill * coeff * (arg ** nu) * kv(nu, arg)
    return full_sill - cov

def fit_variogram_wls(h, gamma, counts, model_func):
    weights = counts / counts.sum()
    def objective(params):
        nugget, sill, range_p, nu = params
        if any(p < 0 for p in params):
            return 1e10
        fitted = model_func(h, nugget, sill, range_p, nu)
        return np.sum(weights * (gamma - fitted) ** 2)
    result = minimize(objective, [0.1, 3.0, 20.0, 1.0],
                      method='Nelder-Mead', options={'maxiter': 5000})
    return result.x

popt = fit_variogram_wls(h_vals[valid], gamma_emp[valid], counts[valid], matern_variogram)
print(f"Matérn fit: nugget={popt[0]:.3f}, sill={popt[1]:.3f}, range={popt[2]:.3f}, nu={popt[3]:.3f}")

# --- Ordinary Kriging ---
def ordinary_kriging(coords, z, new_point, cov_func, **cov_params):
    from scipy.spatial.distance import cdist
    C = cov_func(squareform(pdist(coords)), **cov_params)
    c = cov_func(cdist(new_point, coords).ravel(), **cov_params)
    c0 = cov_func(np.array([0.0]), **cov_params)[0]

    n = len(z)
    C_ext = np.block([[C, np.ones((n, 1))], [np.ones((1, n)), 0]])
    rhs = np.append(c, 1.0)

    try:
        sol = np.linalg.solve(C_ext + 1e-8 * np.eye(n + 1), rhs)
    except np.linalg.LinAlgError:
        sol = np.linalg.lstsq(C_ext, rhs, rcond=None)[0]

    weights = sol[:n]
    pred = weights @ z
    variance = c0 - weights @ c
    return pred, variance

def cov_func(dist, nugget, sill, range_param, nu):
    from scipy.special import kv, gamma as gamma_func
    dist = np.maximum(dist, 1e-10)
    coeff = (2 ** (1 - nu)) / gamma_func(nu)
    arg = np.sqrt(2 * nu) * dist / range_param
    return nugget + sill * coeff * (arg ** nu) * kv(nu, arg)

new_pt = np.array([[50.0, 50.0]])
pred, var = ordinary_kriging(coords, z, new_pt, cov_func,
                              nugget=popt[0], sill=popt[1],
                              range_param=popt[2], nu=popt[3])
print(f"\nOK prediction at (50,50): {pred:.3f} ± {np.sqrt(max(var, 0)):.3f}")

# --- Cross-validation ---
errors, sse = [], []
for i in range(n):
    mask = np.arange(n) != i
    pred_i, var_i = ordinary_kriging(coords[mask], z[mask], coords[i:i+1], cov_func,
                                      nugget=popt[0], sill=popt[1],
                                      range_param=popt[2], nu=popt[3])
    errors.append(z[i] - pred_i)
    sse.append(errors[-1] ** 2)

errors = np.array(errors)
print(f"\nLOOCV Mean Error: {errors.mean():.4f}")
print(f"LOOCV RMSE: {np.sqrt(np.mean(sse)):.4f}")
print(f"LOOCV MAD: {np.mean(np.abs(errors)):.4f}")

Summary