Regionalized Variables and Spatial Estimation: Key Concepts, Stationarity and Sampling
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Regionalized Variables
Topic 1: Regionalized Variables: a function that gives the value of a characteristic "Z" at a point of the studied space "x" that belongs to a random space of dimension R1, R2 or R3. Therefore, the regionalized variable, as it depends on a spatial component, is equal to a spatial variable. This also means that there is a certain dependence between the variable Z and the observation point x.
Characteristics of Regionalized Variables
1. Random and structured aspects: every regionalized variable presents a random aspect and a structured aspect, which depend on what is reflected in the variogram.
2. Support: the volume over which the variable to be studied is measured or considered (point, surface, volume).
3. Geometric field: the area of study in which the experimental data are located.
4. Spatial continuity: the correlation characteristics presented by the variable. That is, there is a relationship between the value measured and the place where it was measured: neighboring and similar data.
5. Anisotropy: the gradient of change of spatial continuity in the different directions of space.
- Isotropic: when the same result is obtained in all directions.
- Anisotropic: when there are changes in the different directions of space.
6. Drift or trend: the directional characteristic of regionalized variables in which they tend to increase or decrease numerically in different directions of space. It is associated with anisotropy.
Stationarity Hypotheses
Stationary hypothesis: F is considered constant with respect to its properties when a translation is applied. It is a hypothesis about the first-order moment or mean, in which this is constant.
2. Second-order moments or covariances are constant and depend only on the distance vector.
Intrinsic hypothesis: it also affects the first two orders, and now we speak of the stationarity of the difference between two variables (increments of the variable).
1. The first-order moment is constant.
2. The second-order moment depends only on h.
Topic 3: Need for Spatial Estimation
We estimate the variable of interest because we cannot always obtain a large set of data for the variable, for economic reasons or due to geolocation constraints.
Thematic map: before applying the method, we must interpret the data of the variable under study through a graphic or topographic representation called a thematic map (TM). We produce a map of the experimental data of the variable.
From experimental spatial data to maps with known continuous values? By means of spatial estimation (SE).
Statement of the Spatial Estimation Problem
We must estimate the value of Z of a variable at a point x, in a given area, from the experimental data taken.
The problem depends on:
- 1. The experimental data: how they are distributed in space, what they are like and what their values are.
- 2. The method: which tool to use to carry out the estimation from the spatial data that exist around it.
- 3. The result: the continuous TMs or maps obtained from this estimation. They will serve to understand and analyze the data.
Characteristics of Sampling
They are studied through technical aspects such as:
- 1. Homogeneity of sampling: measurements must be made under equal conditions for each datum so that they are representative.
- 2. Support of information: the measurement in the sample of the variable in a point, surface or volume form. It modifies the distribution, dispersion and spatial correlation characteristics of the variable for each support.
- 3. Sampling density: refers to the number of samples per unit of area. It is usually given in km2 or as the number of boreholes per unit of area.
- 4. Sampling grid: the spatial arrangement of the sampling.