**4.2.2 Data integration**

280 Remote Sensing of Biomass – Principles and Applications

Fig. 2. Forest resources in Chile and geographical location of the study areas.

The integration of all layers of information was carried out in GIS environment (ArcGIS 9.3RM) and grouped into four classes:


For each of the four study areas a database was generated, which was used subsequently in the geostatistical analysis. Data were treated as point processes or count variables in the bidimensional space. Each set of field data is associated with a specific geographical area, for which all values to a resolution of 16 or 30 meters were considered in a comprehensive manner. Figure 4 shows the general methodological approach and the generic structure of the databases for each area.

From here and for the remaining sections, we considered biomass and all covariates mentioned above as regionalized variables to be able to follow the formality of geostatistical analysis. First, we performed a variogram analysis with the aim of studying the spatial autocorrelation of biomass and its spatial dependence with covariates. The result is a set of spatial models called variograms for the variable of interest (biomass) and the covariates. In all cases, we analyzed the anisotropy of the models, incorporating it explicitly in subsequent interpolations whenever possible. The variograms were used in the spatial estimation of biomass via cokriging. The exploratory analysis and modeling process were performed using IsatisRM geostatistical software and MatlabRM environment.

Geostatistical Estimation of Biomass Stock in Chilean Native Forests and Plantations 283

Once calculated, the experimental variogram is fitted with a theoretical model, which usually consists of a set of basic nested structures with different shapes, sills and/or ranges (Gringarten and Deutsch, 2001). Examples of basic structures include the nugget effect (discontinuous component), as well as the spherical, exponential, cubic, Gaussian and

In the multivariate case, one has to calculate the experimental variogram of each variable (direct variogram), as well as the cross-variograms between each pair of variables, which measure the spatial correlation structure of the set of coregionalized variables. The fitting of a nested structure model is subject to mathematical constraints, reason for which one usually resorts to automatic or semi-automatic fitting algorithms (Wackernagel, 2003).

Kriging aims at estimating the value of the regionalized variable (here, AGB) at a position with no data, as accurately as possible, using the data available in other positions. It is

a. Linearity constraint: The estimator must be a linear combination (i.e., a weighted

b. Unbiasedness constraint: This step expresses that the expected error is zero: if one considers a large number of identical kriging configurations, the average estimation error approaches zero. The absence of bias does not guarantee that errors are small,

c. Optimality constraint: according to the above steps, the estimator is subject to one or more constraints but is not fully specified. The last step consists in minimizing the estimation error variance, which measures the dispersion of the error around its zero mean, therefore the uncertainty in the true unknown value. Intuitively, this restriction means that if a large number of identical kriging configurations are considered, the statistical variance of the estimation errors is as small as possible. In addition to the estimated value, one can also calculate this minimal estimation error variance, known

The weighting of the data depends on the spatial continuity of the regionalized variable, as modeled by the variogram, and on the spatial configuration of the data and location to estimate. In general, closer data receive larger weights than data located far from the target location, but this may be counterbalanced by other effects. For instance, data that are clustered in space tend to convey redundant information and may therefore receive small weights; data located along directions of little spatial continuity (with a small variogram range) may also be underweighted with respect to data located along directions of greater

There exist many variants of kriging, depending on the assumptions about the first-order moment or mean value (in particular, whether it is known or not, constant in space or not) and on the areal support of the value to estimate ("point" support identical to that of the data, or "block" support larger than that of the data) (Chilès and Delfiner, 1999). The multivariate extension of kriging is known as cokriging, which makes use not only of the data of the variable to estimate (primary variable), but also of covariates that are crosscorrelated with this primary variable (Wackernagel et al., 2002). The determination of cokriging weights and of the error variance relies on the set of modeled direct and cross

power models (Chilès and Delfiner, 1999).

structured according to the following steps:

only that they cancel out on average.

**4.2.4 Interpolation by kriging** 

average) of the data.

as the "kriging variance".

variograms (Wackernagel, 2003).

continuity.

Fig. 4. Methodological framework for the integration of information and preparation of files for geostatistical analysis.
