**4.3.2 Variogram analysis**

284 Remote Sensing of Biomass – Principles and Applications

Quite often, the amount of available data is large and using all of the available data is impractical and a selection of the most relevant data is required. One therefore defines a local window or "kriging neighborhood" in which to search for nearby data and to perform local estimation. The shape of the neighborhood should explicitly consider the detected anisotropies, by making the search radius dependent on the variogram range: the larger the range along a direction, the larger the search radius. Typically, the neighborhood is defined by placing an ellipse around the target location, and selecting a given number of data within this ellipse. The justification for such a practice is the so-termed screening effect, according to which the closest data screen out the influence of farthest data (David, 1976; Stein, 2002). Also, to avoid the selection of clustered data that convey redundant information, it is good practice to divide the neighborhood into several angular sectors (e.g., quadrants or octants)

Such a definition of the kriging neighborhood is, however, mainly heuristic and one is usually not aware of which data are really worth being included in the neighborhood, and which data can be discarded without deteriorating the estimation. The optimal neighborhood actually depends on the variogram of the variable of interest, as the screening effect does not occur with every variogram model. Some generic guidelines have been provided by Rivoirard (1987) to validate the choice of a given neighborhood in the

In the multivariate case, the design of the neighborhood is more complex and critical. For instance, the data of a covariate may be screened out by the collocated data of the primary variable or, on the contrary, they may supplement the primary data and provide useful information to improve local estimation. As suggested by recent publications (Rivoirard, 2004; Subramanyam and Pandalai, 2008), the decision to include or not the covariate data should consider the correlation structure of the coregionalized variables and the sampling scheme (in particular, whether or not all the variables are measured at all the sampling

a. Single search: the data points are selected according to their proximity to the target location, irrespective of which variable(s) is (are) informed at these data points. b. Two-part search: the first one is a selection of the nearby data of the primary variable, ignoring the information of covariates; the second one is a selection of the nearby data of the covariates, ignoring the information of the primary variable. The advantage of this strategy is to decouple the search of data according to the nature of the variable (primary or covariate). The disadvantage is that the search and the resolution of the cokriging equations must be carried out as many times as there are variables to

To estimate AGB we used ordinary cokriging (i.e., cokriging with unknown means that are assumed constant at the scale of the neighborhood) considering all the available covariates. Cokriging only works with linearly independent variables, for which there is no colinearity or "redundancy" of information. For this reason, the spectral bands TM1 to TM7 were excluded from the analysis since these variables can be deduced from that of the other variables such as Tasseled Cap components and NDVI. Also, the orientation variable

and to look for data within each sector (Isaaks and Srivastava, 1989).

**4.2.5 Kriging and cokriging neighborhood** 

univariate context.

estimate.

**4.3.1 Selection of variables** 

**4.3 Results** 

locations). Some options include:

While there is much information from the covariates (tens of thousands to millions of records) from which experimental variograms can be calculated in a very detailed way, information is scarcer with the primary variable (AGB) that has only a few hundreds of positions with field data. Because of the limited data available for the country, the inference of the variograms of the primary variable and the cross variograms between this variable and all covariates is difficult. To determine the spatial correlation structure, we chose one of the following alternatives, depending on the case under study:


As an illustration, Figure 5 shows the experimental and modeled variograms for AGB, Brightness (TC1), Greenness (TC2) and Wetness (TC3), along the two identified main directions of anisotropy (north-south N0ºE and east-west N90ºE), for the Pantanillos area. For TC1, TC2 and TC3, the spatial variability appears to be greater along the north-south direction than along the east-west. These direct variograms, together with that of the other covariates and with the cross-variograms (a total of 78 variograms), have been jointly fitted thanks to the algorithm proposed by Goulard and Voltz (1992), using a nugget effect and nested exponential and power basic structures.
