**4.2.4 Interpolation by kriging**

282 Remote Sensing of Biomass – Principles and Applications

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

The values of a regionalized variable such as biomass are not independent, in the sense that the value observed in a site provides information about the values of neighboring sites (i.e. one is more likely to find a high value of biomass near other high values). In the geostatistical interpretation of the regionalized variable this intuitive notion of dependence is described by the formulation of random functions, which model the way the values observed in different sites relate to each other by a multivariate probability distribution. The moment of order 1 of a random function (expected value or mean) involves a single spatial position and does not actually deliver information on spatial dependence. In contrast, the moments of order 2 (especially, the variogram) are defined with the help of two spatial positions, i.e. the smallest set that can be considered to describe the spatial "interaction" between values. These are the moments that give a basic description and

Variogram analysis consists in calculating an experimental variogram from the available data, and subsequently fitting a theoretical variogram model. The experimental variogram measures the mean squared deviation between pairs of data values, as a function of the separation vector (distance and orientation) between the two data. The characteristics of this variogram are related to the characteristics of the regionalized variable (Isaaks and

a. The behavior of the variogram near the origin reflects the short-scale variability of the

b. The increase of the variogram reflects the loss of spatial correlation with the separation vector. It may depend on the direction of this vector, indicating the existence of

c. The range of correlation, for which the variogram reaches a sill (plateau), indicates the

In practice, the anisotropy can be identified by comparing experimental variograms calculated along various directions of space, for example oriented 0°, 45°, 90° and 135° with respect to the x-axis. Often, this test is completed by drawing a "variogram map". When there is isotropy, the experimental variograms in different directions overlap and concentric circles are drawn in the variogram map. Otherwise, one may distinguish geometric anisotropies, in which the variogram map draws concentric ellipses; in such a case, the modeling rests upon the experimental variograms along the main anisotropy axes, which

directions with greater spatial continuity than others (anisotropy).

distance at which two data do no longer have any correlation.

operations of the spatial continuity of the regionalized variable.

Srivastava, 1989; Chilès and Delfiner, 1999), in particular:

for geostatistical analysis.

**4.2.3 Variogram analysis** 

variable.

correspond to the axes of the ellipses.

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 structured according to the following steps:


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 continuity.

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 variograms (Wackernagel, 2003).

Geostatistical Estimation of Biomass Stock in Chilean Native Forests and Plantations 285

corresponds to an angle between 0 and 360 degrees, so the value 0 represents the same information as the value 360. In order to prevent very different values to represent equal or nearly equal angles, we replaced the angles by their cosines and sines. Thus, in addition to the primary variable (AGB), we used the following 11 covariates in all the analyses: altitude (ALT), orientation cosine and sine (ASPECT), slope (SLOPE), normalized difference vegetation index (NDVI), and the six Tasseled Cap components (TC1, TC2, TC3, TC4, TC5

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

a. Use of traditional experimental directional variograms, calculated along the identified

b. Use of omnidirectional variograms, when anisotropy was not detectable for the primary variable (AGB) due to the scarcity of data (Quivolgo and Curacautín case studies). c. Use of centered covariance as a substitute to the variogram (Chilès and Delfiner, 1999) in case the experimental variogram is too erratic (Monte Oscuro case 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

To run cokriging, it is also necessary to define a neighborhood containing the data relevant to the local estimation. Given the very different number of data between the variables (AGB with very few data in comparison with covariates), we decided to use a two-part search:

Cokriging was performed with an *ad hoc* MATLAB routine, since no known commercial software is able to perform cokriging with the above specifications and 11 covariates. The results are estimated values and error variances for AGB. The estimates were made for all the study areas, at the nodes of a grid with cells of 16m × 16m or 30m × 30m, depending on the case study, assuming unknown mean values for all the variables (ordinary cokriging). Since not all the land in each area is covered by vegetation, we subsequently multiplied the estimates and error variances by the fraction of the cells located inside the identified stands, using vector digital layers of their boundaries. Figures 6 and 7 present the field data and

the following alternatives, depending on the case under study:

main directions of anisotropy (Pantanillos case study).

nested exponential and power basic structures.

**4.3.3 Cokriging neighborhood definition and application** 

a. For AGB we considered the 15 closest available data. b. For each covariate, the 50 closest data were considered.

identified stands, as well as the cokriging results for the Pantanillos area.

and TC6).

**4.3.2 Variogram analysis** 
