**1. Introduction**

Soil fertility studies aim to integrate the basic principles of biology, chemistry, and physics, but generally lead to separate interpretations of soil and plants data [1]. Paradoxically, J.B. Boussingeault warned as far as in the 1830s that the balance between nutrients in soil-plant systems was more important than nutrient concentrations taken in isolation [2]. Indeed, the biogeochemical cycles of elements that regulate the dynamics of agroecosystems [3] do not operate independently [4]. However, raw concentrations of individual elements or their log transformation are commonly used to conduct statistical analyses on plant nutrients [5, 6,7], soil fertility indices [8] and C mineralization data [9, 10]. Researchers thus proposed several ratios and stoichiometric rules to relate system's components to each other when monitoring mineralization and immobilization of organic C, N, P and S in soils [4, 11], cations interact‐ ing on soil cation exchange capacity [12], nutrient interactions in plants [13, 14, 15, 16] and carbon uptake by plants [17, 18].

Different approaches have been elaborated to describe nutrient balances in soils. The nu‐ trient intensity and balance concept (NIBC) computes ionic balances in soil water extracts [19, 20]. The basic cation saturation ratio (BCSR) concept hypothesizes that cations and acidi‐ ty exchanging on soil cation exchange capacity (CEC) can be optimized for crop growth [12]. However, the BCSR has been criticized for its elusive definition of 'ideal' cationic ratios [21, 22]. In plant nutrition, [23] were the first to represent geometrically interactions between nu‐ trients by a ternary diagram where one nutrient can be computed by difference between 100% and the sum of the other two. As a result, there are two degrees of freedom in a terna‐ ry diagram. One may also derive three dual ratios from K, Ca, and Mg but only two ratios are linearly independent because, for example, K/Mg can be computed from K/Ca × Ca/Mg

© 2012 Parent et al.; licensee InTech. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. © 2012 Parent et al.; licensee InTech. This is a paper distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

and is thus redundant. Therefore, a ratio approach conveys *D*-1 degrees of freedom or line‐ arly independent balances for a *D*-part composition [24].

log-ratio (*ilr*) transformation developed by [27]. In the literature, the nutrient balance often refers to a nutrient budget that measures the depletion or accumulation of a given nutrient in soils [28], implying exchange between compartments of some whole. In this chapter, nu‐ trient balance is defined as dual or multiple log ratios between nutrients, implying balance

Nutrient Balance as Paradigm of Soil and Plant Chemometrics

http://dx.doi.org/10.5772/53343

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The aim of this chapter is to introduce the reader to the balance concept as applied to soil fertility studies. The first section of this chapter presents the theory common to the three subsequent subjects, which are cationic balance in soils, plant nutrient signatures and miner‐ alization of organic residues. It is suggested that the reader gets familiar with the theory be‐

Because a change in any proportion of a whole reverberates on at least one other proportion, proportions of components of a closed sum (100%) are interdependent. Therefore, a compo‐ sitional vector is intrinsically multivariate: its components cannot be analyzed and interpret‐ ed without relating them to each other [32,33]. Compositional data (CoDa) induce numerical biases, such as self-redundancy (one component is computable by difference between the constrained sum of the whole and the sum of other components), non-normal distribution (the Gaussian curve may range below 0 or beyond 100% which is conceptually meaningless) and scale dependency (correlations depend on measurement scale). Redundancy can be con‐ trolled by carefully removing the extra degree of freedom in the *D*-part composition. Scale dependency is controlled by ratioing components after setting the same scale (e.g. fresh mass, dry mass or organic mass basis) or unit of measurement (e.g. mg kg-1, g dm-3

kg-1, etc.) across components. Compositional datasets constrained to a closed space between 0 and 100% are amenable to normality tests after projecting them into a real space using log-

One of the log ratio transformations is the centered log ratio (*clr*) developed by [26]. The *clr* is a log ratio contrast between the concentration of any nutrient and the geometric mean across the compositional vector. [34] used the *clr* to convert DRIS into Compositional Nu‐ trient Diagnosis (CND-*clr*), hence correcting inherent biases generated by DRIS. [35] and [36] modeled the time change of ion activities in soils and nutrient solutions using *clr*. However, because *clr* generates a singular matrix (the *clr* variates sum up to 0), one *clr* value should be removed (e.g. that of the filling value) in multivariate analysis. In addition, outliers may af‐ fect considerably log ratios [32]. The diagnostic power of CND-*clr* is decreased by large var‐ iations in nutrient levels (e.g. Cu, Zn, Mn contamination by fungicides) that affect the geometric means across concentrations. Nevertheless, the *clr* transformation is useful to con‐

The additive log ratio or *alr* [26] computed as ln(x/xD) is the ratio between any component x and a reference component xD. [17] used nitrogen as reference component (N=100%) to pro‐ duce a stoichiometric N:P:K:Ca:Mg rule for adjusting nutrient needs of tree seedlings. If a

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between components of the same whole.

fore browsing through the subject of interest.

duct exploratory analyses on compositional data [37].

**2. Theory of CoDa**

ratio transformations.

In contrast, there are *D*×(*D*-1)/2 dual ratios such as the K/Mg ratio and *D*×(*D*-1)²/2 two-com‐ ponent amalgamated ratios such as the K/(Ca+Mg) ratio that can be derived from a *D*-part composition. Most information on dual and two-component amalgamated ratios is thus re‐ dundant and the dataset is artificially inflated. In Figure 1, the number of (a) dual and (b) two-component amalgamated ratios is plotted against the number of components. With 10 components, one may compute up to 45 dual and 405 two-component amalgamated ratios, hence generating a "redundancy bubble" that inflates exponentially above *D*. [25] elaborat‐ ed the Diagnosis and Recommendation Integrated System (DRIS) to synthesize the *D*× (*D*-1)/2 dual ratios into *D* nutrient indices adding up to zero; therefore, there is still one re‐ dundant index closing the system to zero and computable from other indices. Applying Ockham's razor law of parsimony to compositional data, nine degrees of freedom suffice to fully describe a 10-part composition without bias [24].

**Figure 1.** Number of (a) dual and (b) two-component amalgamated ratios illustrating the redundancy bubble.

To solve problems related to nutrient diagnosis in soil and plant sciences, one must first rec‐ ognize that soil and plant analytical data are most often compositional, i.e. strictly positive data (concentrations, proportions) related to each other and bounded to some whole [26]. Compositional data have special numerical properties that may lead to wrong inferences if not transformed properly. Log-ratio transformations have been developed to avoid numeri‐ cal biases [26, 29, 30, 31]. The balance concept presented in this chapter is based on log ratios or contrasts. Balances are computed rather simply from compositions using the isometric log-ratio (*ilr*) transformation developed by [27]. In the literature, the nutrient balance often refers to a nutrient budget that measures the depletion or accumulation of a given nutrient in soils [28], implying exchange between compartments of some whole. In this chapter, nu‐ trient balance is defined as dual or multiple log ratios between nutrients, implying balance between components of the same whole.

The aim of this chapter is to introduce the reader to the balance concept as applied to soil fertility studies. The first section of this chapter presents the theory common to the three subsequent subjects, which are cationic balance in soils, plant nutrient signatures and miner‐ alization of organic residues. It is suggested that the reader gets familiar with the theory be‐ fore browsing through the subject of interest.
