Semi-variogram
Definition
A semi-variogram is a fundamental tool in geostatistics used to describe the spatial dependence of variables. It plots the variance of the difference between sample values against the distance separating those samples. The semi-variogram is crucial for determining how data points interact spatially, providing insights into spatial patterns and the level of spatial correlation among samples. This is key for spatial interpolation methods like kriging, which leverage these spatial relationships to predict values at unsampled locations.
What is a Semi-variogram?
The semi-variogram is a function that quantifies how similar or dissimilar values are at varying distances in space. The x-axis of the plot generally represents the distance (termed lag distance) between pairs of sampled locations, while the y-axis represents the semi-variance, which measures the degree of dissimilarity between data values at those locations. As distance increases, the semi-variance typically increases, indicating reduced similarity between points.
Within the semi-variogram, several core components are recognized:
- Nugget: Represents the variance at zero distance, indicating measurement error or spatial variation at micro scales.
- Range: The distance at which the semi-variance levels off, beyond which data points are no longer spatially correlated.
- Sill: The value that the semi-variance reaches at the range distance, showing the limit of spatial correlation.
The semi-variogram is central in developing spatial prediction models because it provides the necessary statistical model to inform the kriging process, allowing for accurate predictions and uncertainty assessments.
FAQs
What is the purpose of a semi-variogram in geostatistics?
The purpose of a semi-variogram in geostatistics is to analyze and model the spatial correlation structure of data. It helps in understanding how data values change with distance and informs interpolation methods like kriging used for generating spatial predictions.
How is a semi-variogram constructed?
A semi-variogram is constructed by calculating the semi-variance for pairs of sample points at various distances. This involves taking the squared differences of sample pairs and averaging them for each distance class, then plotting these averages against the corresponding distances to form the semi-variogram.
What is the difference between a semi-variogram and a variogram?
A semi-variogram and a variogram essentially describe the same spatial relationships, but a variogram includes the factor of two in its formula (( \gamma(h) = \frac{1}{2} Var(Z(x) - Z(x+h)) )). In practice, the terms are often used interchangeably, with semi-variogram being more common in geostatistical discussions.
How is a semi-variogram used in kriging?
In kriging, a semi-variogram is used to weight the influence of sampled locations on the estimation of unsampled points. The spatial correlation structure modeled by the semi-variogram helps determine how much each sampled point should contribute to a prediction, based on the distance and arrangement of sample locations.
