Kriging

Key Takeaways

• The kriging system of equations determines the optimal weights assigned to each surrounding data point by solving a set of linear equations derived from the variogram model and the requirement that the estimates be unbiased: for ordinary kriging (the most commonly used variant, which estimates the unknown mean of the field locally from the data in the search neighborhood rather than assuming a globally known mean), the kriging weights must sum to 1 (the unbiasedness constraint), and the optimal weights are computed by solving a matrix equation whose elements are the variogram values between all pairs of data points and between each data point and the estimation location; data points close to the estimation location receive higher weight than distant points, but the kriging weights also account for data redundancy: two closely spaced data points carry less independent information than two widely spaced points, and the kriging system automatically downweights closely spaced (redundant) data pairs relative to isolated data points; this spatial declustering property of kriging is one of its major advantages over simpler distance-based interpolation methods (inverse distance weighting) that cannot account for data clustering.
• The kriging variance (sigma-squared-k) measures the uncertainty of the kriging estimate at each location as a function of the data configuration and the variogram, providing a spatially variable uncertainty map that complements the kriging estimate map: the kriging variance is zero at data locations (where the estimate equals the measured value exactly, with no uncertainty) and increases with distance from data points (where less information is available and the estimate is less reliable); the square root of the kriging variance (the kriging standard deviation) can be used to compute confidence intervals for the estimates at each location, assuming the spatial random field has a Gaussian distribution; however, the kriging variance depends only on the data configuration and variogram model, not on the data values themselves, meaning that a smoothly distributed data set in a high-variogram-nugget, short-range field will have the same kriging variance pattern as a uniformly sampled field even if the actual variability of the data values is very different; this property is both a strength (uncertainty is objectively defined by data geometry) and a limitation (extreme data values near a location do not increase the local uncertainty estimate beyond what the geometry implies).
• Kriging variants address different assumptions about the spatial trend and statistical properties of the property being estimated: ordinary kriging (OK, assumes local stationarity of the mean but does not require the mean to be known globally), simple kriging (SK, assumes the mean is known and constant everywhere, used in sequential Gaussian simulation), universal kriging (UK, also called kriging with a trend or external drift, incorporates a spatially varying deterministic trend component such as depth-dependent compaction or seismically derived acoustic impedance as an external variable), and co-kriging (incorporates secondary variables that are spatially correlated with the primary property of interest to improve estimation accuracy by leveraging the denser sampling of secondary data); in petroleum applications, co-kriging with seismic attributes (acoustic impedance from seismic inversion, amplitude from seismic reflection, or attributes from spectral decomposition) as the secondary variable is a powerful approach for improving porosity or net-to-gross estimation between wells where seismic data provides dense lateral coverage that constrains the interpolation.
• The smoothing effect of kriging is a fundamental limitation that makes kriged property maps appropriate for some applications but inappropriate for others: kriging produces smooth, best-estimate maps that minimize the overall estimation error, but the local variability of the property is underrepresented in the kriged result because kriging weights average the surrounding data values and cannot reproduce values more extreme than the data themselves at unsampled locations; the kriged porosity map of a heterogeneous reservoir may show correct average properties at each location but will not reproduce the high-porosity channel sands or low-porosity cemented zones that control reservoir connectivity and permeability; for applications requiring a realistic representation of property variability (reservoir simulation, uncertainty quantification, or connectivity analysis), stochastic simulation methods (sequential Gaussian simulation, sequential indicator simulation) that reproduce the variogram statistics while generating extreme values consistent with the data histogram are preferred over kriging; kriging is appropriate when the goal is a single best-estimate map for structural interpretation, volumetric calculation, or input to a deterministic reservoir simulation history-match model.
• Indicator kriging (IK) adapts the kriging framework to categorical or non-Gaussian data by transforming the original continuous data into binary indicator variables (1 if the value exceeds a threshold, 0 if it does not) and kriging the indicator values to produce probability maps of exceeding each threshold: the indicator approach is used for estimating the probability of sand vs. shale at each location (using lithological indicators from well log facies analysis), the probability of exceeding a porosity cutoff, or the probability of a reservoir property being in a specified range; multiple threshold indicator kriging at several cumulative probability levels can reconstruct the complete conditional probability distribution of the property at each location, providing a non-parametric spatial uncertainty model that makes no assumption about the shape of the distribution (unlike ordinary kriging, which implicitly assumes Gaussian statistics); indicator kriging is widely used in reservoir characterization for facies probability mapping and in reserve estimation for producing probability distributions of reservoir property volumes that feed into probabilistic reserve calculations.