**3. Cationic balances in tropical soils**

#### **3.1. Sequential binary partition**

The percentage base saturation is the proportion of soil cation exchange capacity (CEC) oc‐ cupied by a given cation. The soil compositional vector is defined as follows [12]:

$$\mathcal{S}^{\mathsf{A}} = \mathcal{C}\{\mathcal{K}, \ \mathsf{Ca}, \ \mathsf{Mg}, \ \mathsf{H} \star \mathsf{Al}\}\tag{7}$$

As illustrated in Figure 7, the first contrast, [K | Ca, Mg, H+Al], balances the K against diva‐ lent cations and acidity to enable adjusting the K fertilization to soil basic acid-base condi‐ tions as modified by liming.

The second contrast [Ca, Mg | H+Al] is the acid-base contrast for determining lime require‐ ments while the [Ca | Mg] balance reflects the Ca:Mg ratio in soils adjustable by the liming materials. Alternative SBPs could also be elaborated such as [K, Ca, Mg | (H+Al)], [K | Ca, Mg] and [Ca | Mg] balances that reflects the BCSR model of [12]. The selected sequential binary partition for cationic balances is presented in Table 2.

For example, if a soil contains 2.9 mmolc K dm-3, 20 mmolc Ca dm-3, 5 mmolc Mg dm-3, and 23 mmolc H+Al dm-3. Cationic balances are computed as follows:

$$
\binom{1}{1}\,\mathrm{L}\,K\,\mid\,\mathrm{Ca},\,\,Mg\_{\mathrm{e}}\,\,H+\,\mathrm{Al}\,\bigg|=\sqrt{\frac{1\times3}{1+3}}\ln\left(\frac{2.9}{\sqrt{20\times5,\ \mathrm{x}\,23}}\right) = -1.312;
$$

$$
\binom{2}{2}\,\mathrm{K}\,\,\,^{\mid}\,\mathrm{Ca},\,\,Mg\_{\mathrm{g}}\,\bigg|=\sqrt{\frac{1\times2}{1+2}}\ln\left(\frac{2.9}{\sqrt{20\times5}}\right) = -1.011;\,\,\mathrm{and}
$$

$$
\binom{3}{3}\,\,\mathrm{Ca}\,\,\,^{\mid}\,\mathrm{Mg}\,\bigg|=\sqrt{\frac{1\times1}{1+1}}\ln\left(\frac{20}{5}\right) = 0.980.
$$

Note that the K fertilization would depend on soil acidity as well as levels of exchangeable Ca and Mg in the soil. We thus expect the K index and the K balance to be similarly related to fruit yield if the *ceteris paribus* assumption applies to exchangeable Ca, Mg, and acidity in this soil-plant system.

experiment lasted 3 yr. The N treatments in the 1st year were 0, 30, 60, 120, 180, 240 and 300 g N tree-1 supplemented with 52 g P tree-1 and 52 g K tree-1. The initial N rates were doubled and tripled in the 2nd and 3rd years, respectively. The initial P and K doses were doubled the 2nd year. The 3rd year, rates were 240 g P2O5 tree-1 and 360 g K2O tree-1. Fertil‐ izers were ammonium nitrate (34% N), simple superphosphate (8.7% P) and potassium chloride (50% K). In the K trial, K was added as KCl at rates of 0, 25, 50, 100, 150 and 200 and 250 g K tree-1 the 1st year and supplemented with 120 g N tree-1 as ammonium sul‐ fate (20% N) and 52 g P tree-1 as triple superphosphate (19% P). The N, P, and K rates were doubled in the 2nd year. The K rates were tripled the 3rd year and supplemented with 360 g N tree-1 and 105 g P tree-1. The acidifying ammonium fertilizers may increase exchangeable acidity in both trials. The fertilizers were broadcast around the tree 40 cm from crown projection. Each plot comprised four trees each covering an area of 7 m x 5 m, for a total of 286 trees ha-1. The experimental setup was a randomized block design with four replications. Fresh fruit yields were measured 1-3 times wk-1 from January to

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Soils were sampled annually after harvest at four locations per tree in the 0-20 cm and 20-40 cm layers where most of the root system is located, then composited per plot. Soil samples were air dried and analyzed for K, Ca, Mg and (H + Al) [61]. The K, Ca and Mg were extract‐ ed using exchange resins, quantified by atomic absorption spectrophotometry and reported as mmolc dm-3. Exchangeable acidity (H+Al) was quantified by the SMP pH buffer method [62] and computed using the equation of [63] to convert buffer pH into mmolc (H+Al) dm-3

Cation exchange capacity (CEC) was computed as the sum of cationic species. Assuming a

As shown by scatter and ternary diagrams (Figure 8), the large ellipses, that represent the distribution of cationic balances in the 0-20 and 20-40 cm layers, overlapped. However, the small ellipses (Figure 8) representing the confidence region about means differed signifi‐ cantly. The [K │ Ca, Mg, H+Al] balance was higher in the 0-20 cm layer, indicating that more K accumulated in the surface layer as a result of surface K fertilizer applications.

Soil test K and cationic balances were averaged between the beginning and the end of the growing season to represent average soil conditions. The soil indices were related to fresh fruit yield (Figures 9a and 9b). In Figure 9, data are means of 4 replicates and bars are least

(*H* + *Al*)=10exp (7.76 + 1.053*pH SMP*), R² =0.98 (8)

June, starting approximately 97 d after fruit set.

soil bulk density of 1 kg dm-3, CEC averaged 5.4 cmolc kg-1.

*3.3.1. Influence of the K fertilization on cationic balances in soil*

as follows:

**3.3. Results**

significant differences.

**Figure 7.** The cationic balances in soils can be designed to facilitate K fertilization and lime management. Each bal‐ ance between two groups of ions is computed as isometric log ratio.


**Table 2.** Sequential binary partition of soil cationic data

#### **3.2. Datasets**

Changes in soil cationic balances were monitored in N and K fertilizer trials established on an epieutrophic and endodystrophic soil (Red-Yellow Oxisol) [60] at São Carlos (São Paulo, Brazil). One year old plants of 'Paluma' guava (*Psidium guajava*) were planted. The experiment lasted 3 yr. The N treatments in the 1st year were 0, 30, 60, 120, 180, 240 and 300 g N tree-1 supplemented with 52 g P tree-1 and 52 g K tree-1. The initial N rates were doubled and tripled in the 2nd and 3rd years, respectively. The initial P and K doses were doubled the 2nd year. The 3rd year, rates were 240 g P2O5 tree-1 and 360 g K2O tree-1. Fertil‐ izers were ammonium nitrate (34% N), simple superphosphate (8.7% P) and potassium chloride (50% K). In the K trial, K was added as KCl at rates of 0, 25, 50, 100, 150 and 200 and 250 g K tree-1 the 1st year and supplemented with 120 g N tree-1 as ammonium sul‐ fate (20% N) and 52 g P tree-1 as triple superphosphate (19% P). The N, P, and K rates were doubled in the 2nd year. The K rates were tripled the 3rd year and supplemented with 360 g N tree-1 and 105 g P tree-1. The acidifying ammonium fertilizers may increase exchangeable acidity in both trials. The fertilizers were broadcast around the tree 40 cm from crown projection. Each plot comprised four trees each covering an area of 7 m x 5 m, for a total of 286 trees ha-1. The experimental setup was a randomized block design with four replications. Fresh fruit yields were measured 1-3 times wk-1 from January to June, starting approximately 97 d after fruit set.

Soils were sampled annually after harvest at four locations per tree in the 0-20 cm and 20-40 cm layers where most of the root system is located, then composited per plot. Soil samples were air dried and analyzed for K, Ca, Mg and (H + Al) [61]. The K, Ca and Mg were extract‐ ed using exchange resins, quantified by atomic absorption spectrophotometry and reported as mmolc dm-3. Exchangeable acidity (H+Al) was quantified by the SMP pH buffer method [62] and computed using the equation of [63] to convert buffer pH into mmolc (H+Al) dm-3 as follows:

$$\Pr(H+Al) = 10 \exp\left\{7.76 + 1.053pH\_{\text{SMP}}\right\} \text{ } \text{ R}^2 = 0.98 \tag{8}$$

Cation exchange capacity (CEC) was computed as the sum of cationic species. Assuming a soil bulk density of 1 kg dm-3, CEC averaged 5.4 cmolc kg-1.

#### **3.3. Results**

**Figure 7.** The cationic balances in soils can be designed to facilitate K fertilization and lime management. Each bal‐

Changes in soil cationic balances were monitored in N and K fertilizer trials established on an epieutrophic and endodystrophic soil (Red-Yellow Oxisol) [60] at São Carlos (São Paulo, Brazil). One year old plants of 'Paluma' guava (*Psidium guajava*) were planted. The

<sup>1</sup> <sup>+</sup> <sup>3</sup> *ln*( *<sup>K</sup>*

<sup>1</sup> <sup>+</sup> <sup>2</sup> *ln*( *<sup>K</sup>*

<sup>1</sup> <sup>+</sup> <sup>1</sup> *ln*( *Ca Mg* )

*CaxMg*, *<sup>x</sup>*(*<sup>H</sup>* <sup>+</sup> *<sup>A</sup>* <sup>3</sup> *<sup>l</sup>*) )

*CaxMg* )

ance between two groups of ions is computed as isometric log ratio.

K Ca Mg H+Al

**Table 2.** Sequential binary partition of soil cationic data

**3.2. Datasets**

94 Soil Fertility

**Partition Cationic balances r s** *ilr formulation*

1 1 -1 -1 -1 3 1 1*x*3

2 1 -1 -1 0 1 2 1*x*2

3 0 1 -1 0 1 1 1*x*1

#### *3.3.1. Influence of the K fertilization on cationic balances in soil*

As shown by scatter and ternary diagrams (Figure 8), the large ellipses, that represent the distribution of cationic balances in the 0-20 and 20-40 cm layers, overlapped. However, the small ellipses (Figure 8) representing the confidence region about means differed signifi‐ cantly. The [K │ Ca, Mg, H+Al] balance was higher in the 0-20 cm layer, indicating that more K accumulated in the surface layer as a result of surface K fertilizer applications.

Soil test K and cationic balances were averaged between the beginning and the end of the growing season to represent average soil conditions. The soil indices were related to fresh fruit yield (Figures 9a and 9b). In Figure 9, data are means of 4 replicates and bars are least significant differences.

(a)

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(b)

**Figure 9.** Exchangeable K (a), the [K | Ca,Mg,H+Al] balance (b) and fruit yield increased with added K. Other cationic

The Cate-Nelson partitioning of the relationship between guava fresh fruit yield and either soil K level or the [K | Ca, Mg, (H+Al)] balance across the combined N and K fertilizer experiments indicates that the K level index classified two specimens as TN compared to four for the K bal‐ ance index (Figure 10). The graphical representation of this soil-plant relationship indicates di‐

agnostic advantage to using the K nutrient balance in rather than the K concentration.

*3.3.2. Critical soil K concentration and balance in the N and K trials*

balances did not change markedly.

**Figure 8.** The K accumulated relatively more on the 0-20 cm layer following three years of K fertilization as shown by (a) two orthonormal cationic balances and (b) a K-Ca-Mg ternary diagram.

**Figure 9.** Exchangeable K (a), the [K | Ca,Mg,H+Al] balance (b) and fruit yield increased with added K. Other cationic balances did not change markedly.

#### *3.3.2. Critical soil K concentration and balance in the N and K trials*

(a)

96 Soil Fertility

(b)

**Figure 8.** The K accumulated relatively more on the 0-20 cm layer following three years of K fertilization as shown by

(a) two orthonormal cationic balances and (b) a K-Ca-Mg ternary diagram.

The Cate-Nelson partitioning of the relationship between guava fresh fruit yield and either soil K level or the [K | Ca, Mg, (H+Al)] balance across the combined N and K fertilizer experiments indicates that the K level index classified two specimens as TN compared to four for the K bal‐ ance index (Figure 10). The graphical representation of this soil-plant relationship indicates di‐ agnostic advantage to using the K nutrient balance in rather than the K concentration.

The sensitivity, specificity, PPV and NPV criteria are presented in Table 3. We expect perform‐ ance criteria to be at least 80%. Low specificity indicates that some interactions with K leading to high yield, possibly involving Ca and Mg, have been ignored. Apparently, the *ceteris paribus* assumption did not apply to this study. The fact that the balance allows to adjust the K to other cationic species may account for failure to meet the *ceteris paribus* assumption.

**Soil K index Sensitivity =**

**4.1. Sequential binary partition**

for conducting statistical analysis:

**•** N with S, P, K, Ca, Mg, Fe, Mn, Zn, and Cu;

**•** P with N, K, Ca, Mg, B, Mo, Cu, Fe, Mn, Al, and Zn;

**•** K with N, P, Ca, Mg, Na, B, Mn, Mo, and Zn;

**•** Ca with N, K, Mg, Na, Cu, Fe, Mn, Ni, and Zn;

**•** NH4 with K, Ca, and Mg;

**•** S with N, P, Fe, Mn, Mo;

**•** Cl with N and S;

**TP/(TP+FN)**

**Specificity = TN/(TN+FP) Positive**

K level 100.0 66.7 91.7 100.0

K balance 100.0 80.0 90.0 100.0

Plant nutrients are classified as essential macronutrients measured in % (N, S, P, Mg, Ca, K, Cl), essential micronutrients measured in mg kg-1 (Mn, Cu, Zn, Mo, B) and beneficial nu‐ trients generally measured in mg or µg kg-1 but occasionally in % (Si, Na, Co, Ni, Se, Al, I, V) [64, 65, 15]. The plant ionome is defined as elemental tissue composition as related to the genome [66]. A subcomposition of plant ionome could be defined by the following simplex

Where *Fv* is the filling value between 1000 g kg-1 and the sum of analytical data and *D* = 15, the total number of components including *Fv*. An SBP scheme can be elaborated based on well documented roles and stoichiometric rules provided by [17, 14, 12], who reported a

large number of dual and multiple nutrient interactions in plants such as: **•** Macronutrients have a stoichiometric relationship with carbon uptake;

*<sup>S</sup> <sup>D</sup>* <sup>=</sup>(*C*, *<sup>N</sup>* , *<sup>P</sup>*, *<sup>K</sup>*, *Ca*, *Mg*, *<sup>B</sup>*, *<sup>S</sup>*, *Cl*, *Cu*, *Zn*, *Mn*, *Fe*, Mo, *<sup>F</sup> <sup>v</sup>*) (9)

**Table 3.** Performance of K indices in terms of sensitivity, specificity, PPV and NPV

**4. Multi-element Balances in plant nutrition**

**%**

**predictive value = PPV=TP/(TP+FP)**

Nutrient Balance as Paradigm of Soil and Plant Chemometrics

**Negative predictive value = NPV=TN/(TN+FN)** 99

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**Figure 10.** Cate-Nelson partitioning of the relationship between guava fresh fruit yield. critical values were (a) 1.2 mmolc K dm-3. TN = 2; FN = 0; TP = 11; FP = 1 and (b) -2.07. TN = 4; FN = 0; TP = 9; FP = 1.


**Table 3.** Performance of K indices in terms of sensitivity, specificity, PPV and NPV
