Geostatistics Study Guide

📖 Core Concepts Geostatistics – statistical analysis of spatial (or spatiotemporal) data; treats values at locations as random variables. Random function theory – the variable of interest \(Z(\mathbf{x})\) is a random variable at unsampled sites; its behavior is described by probability distributions. Spatial continuity – nearby locations tend to have similar values; quantified with a variogram (semi‑variance vs lag). Stationarity – statistical properties (mean, variance, covariance) are constant over the study area; a single variogram model assumes stationarity. Estimation vs. Simulation – Estimation predicts a single best value (e.g., kriging mean). Simulation draws many possible realizations to represent uncertainty. Kriging – optimal linear unbiased estimator that uses the covariance/variogram to weight nearby data. Bayesian geostatistics – treats kriging as a Gaussian‑process posterior update after incorporating new data. Covariance function – describes how two locations co‑vary as a function of distance; the variogram is derived from it. Key variogram parameters – nugget (value at zero lag, measurement error/microscale), sill (plateau value = total variance), range (lag where plateau is reached → points become essentially independent). Training image – a realistic pattern used in multiple‑point simulation to guide the generation of spatial realizations. --- 📌 Must Remember Variogram formula: \(\gamma(h) = \frac{1}{2}E[(Z(\mathbf{x})-Z(\mathbf{x}+h))^{2}]\). Nugget‑sill‑range fully describe a simple variogram model. Kriging weights are solved from the system \(\mathbf{C}\mathbf{w} = \mathbf{c}\) where \(\mathbf{C}\) is the covariance matrix of data points and \(\mathbf{c}\) the covariance vector between data and target. Stationarity assumption ⇒ one variogram model for the whole domain; violation → use local variograms or non‑stationary methods. Estimation goal → use expectation (mean), median, or mode of the conditional CDF. Simulation goal → generate many realizations sampling the full joint PDF; each realization respects the variogram (or training image). Tobler’s first law: “Near things are more related than distant things” → foundation of spatial continuity. --- 🔄 Key Processes Variogram Construction Compute pairwise semi‑variances for a set of lag distances. Plot \(\gamma(h)\) vs. \(h\). Fit a parametric model (spherical, exponential, Gaussian) extracting nugget, sill, range. Ordinary Kriging Estimation Choose a variogram model. Build covariance matrix \(\mathbf{C}\) for neighboring data. Assemble covariance vector \(\mathbf{c}\) between data and target location. Solve \(\mathbf{C}\mathbf{w} = \mathbf{c}\) for weights \(\mathbf{w}\). Estimate: \(\hat{Z}(\mathbf{x}0) = \sum{i} wi Z(\mathbf{x}i)\). Bayesian Updating (Gaussian Process) Prior: mean \(\mu\) and covariance from variogram. Likelihood: observed data with measurement error (nugget). Posterior mean = kriging estimator; posterior covariance = kriging variance. Multiple‑Point Simulation Select a training image representing the desired pattern. Scan the image to collect conditional probabilities for point neighborhoods. Sequentially assign values to grid nodes, sampling from the conditional distribution guided by the training image. --- 🔍 Key Comparisons Kriging vs. Simple Inverse Distance Weighting (IDW) Kriging: weights derived from statistical model (covariance); provides estimation variance. IDW: deterministic, weights = \(1/d^{p}\); no variance estimate, assumes isotropy implicitly. Estimation vs. Simulation Estimation: single “best guess”; ignores full uncertainty. Simulation: many possible maps; captures uncertainty and spatial patterns. Stationary vs. Non‑stationary Models Stationary: one variogram for whole area; easier but may misrepresent trends. Non‑stationary: local variograms, trend‑removal, or kernel‑based covariance; more flexible. Parametric vs. Non‑parametric Variogram Models Parametric: spherical, exponential – few parameters, easy to fit. Non‑parametric: empirical variogram, multiple‑point simulation – captures complex structures but harder to extrapolate. --- ⚠️ Common Misunderstandings Nugget = measurement error only – it also captures microscale variability that the sampling cannot resolve. “Kriging always gives the true value” – kriging provides the best linear unbiased estimate under the assumed model; model misspecification leads to bias. Range = distance beyond which points are completely unrelated – technically, correlation is very low beyond the range, not zero. Variogram plateau = sill always equals total variance – only true for second‑order stationary processes; trends inflate apparent sill. --- 🧠 Mental Models / Intuition “Spatial glue” – imagine the variogram as a rubber band: the tighter (small nugget, long range) the band, the more neighboring points pull each other toward similar values. Kriging as weighted averaging with “confidence” – points closer (higher covariance) get larger weights; the kriging variance tells you how “confident” the estimate is. Simulation = painting many possible pictures – each realization respects the same statistical “brush strokes” (variogram or training image) but adds random variation to show uncertainty. --- 🚩 Exceptions & Edge Cases Strong anisotropy – direction‑dependent continuity requires anisotropic variogram models (different ranges per direction). Sparse data – kriging may revert to the global mean; simulation can still produce plausible patterns if a training image is used. Non‑Gaussian data – ordinary kriging assumes Gaussianity; for skewed or categorical data use indicator kriging or multiple‑point simulation. --- 📍 When to Use Which Use ordinary kriging when: data are roughly Gaussian, stationarity reasonable, you need a single best estimate + variance. Use indicator or cokriging when: variables are categorical or you have secondary correlated variables. Use Bayesian Gaussian‑process kriging when: you want a full posterior distribution and can incorporate prior information. Use multiple‑point simulation when: the spatial pattern is complex (e.g., channels, facies) and a training image is available. Use simple IDW only for quick, rough estimates or when a variogram cannot be reliably estimated. --- 👀 Patterns to Recognize Variogram shape → continuity type: Spherical: clear range, abrupt plateau – common for soils. Exponential: gradual approach to sill – indicates smoother decay of correlation. Gaussian: very smooth, long‑range correlation. Flat variogram at small lags → strong nugget (high measurement error or microscale noise). Kriging variance map low near data, high far away – typical “confidence map” pattern. --- 🗂️ Exam Traps Choosing “range = distance where variogram = sill” – some textbooks define range at 95 % of sill; watch the exact definition given in the question. Confusing semi‑variance with variance – semi‑variance is half the average squared difference, not the overall variance. Assuming kriging always reduces variance – if the variogram model is wrong, kriging variance can be misleadingly low. Selecting the wrong variogram model for anisotropy – an isotropic model will be penalized in a question that mentions direction‑dependent continuity. Mixing up training image with variogram – a training image is a pattern source for multiple‑point simulation, not a covariance model. ---