引言

本复现项目实现了多种热带酸性土壤石灰需要量计算公式，核心目标包括：

1. 实现 Kamprath、Cochrane、NuMaSS 及巴西 V 法的石灰推荐逻辑
2. 生成与原始文献一致的铝饱和度下降曲线、预测值与实际值对比图
3. 提供单位转换（meq/100g ↔︎ 吨/公顷）及石灰材料等效性调整功能

在开始之前，需要注意以下几点：

• 所有石灰用量均以 meq/100g 土壤表示，等同于 cmol_c/kg。每面积石灰用量（如吨/公顷）取决于单位面积土壤重量，而后者取决于土壤容重和石灰掺入深度。参见 convert() 函数了解如何在不同单位之间转换。

• 石灰用量指纯细质地方解石石灰（即 100% CaCO₃）的用量。 参见 CCE() 函数了解与其他石灰材料的等效性以及如何根据粒径调整。

library(limer)

This vignette provides a review of lime recommendation formulas and their usage. The review focuses on formulas for tropical acid soils that can be used in ex-ante analyses, with readily available soil data, and without additional soil tests.

Some considerations before we dive into the formulas:

• All lime rates are expressed in meq/100g of soil, which is the same as cmol_c/kg. Lime rate per area (e.g., tons per hectare) depends on soil weight per area, which in turn depends on soil bulk density and lime incorporation depth (soil depth). See function convert() for how to convert between lime rate units.

• Lime rates refers to the amount of pure fine-textured calcitic limestone (i.e., 100% CaCO₃). See function CCE() for equivalencies with other liming materials and how to adjust for particle size.

Exchangeable Aluminum formulas

This set of formulas calculate the lime rate required to reduce exchangeable Aluminum (or acidity, see below) levels to a level that the crop can tolerate. They rely on the assumption that aluminum toxicity is the main factor that needs to be considered in acid soils, and that crops can tolerate certain levels of exchangeable Aluminum without a great impact on crop yields.

Aluminum toxicity is a function of the of the aluminum saturation (usually expressed as %), which is the exchangeable Al divided by effective cation exchange capacity (ECEC), and the main target of these formulas

Importantly, despite most formulas use exchangeable Aluminum, we recommend calculating lime rates based on exchangeable acidity. See the Appendix at the end of the vignette for a discussion about this.

method = "ka": Kamprath (1970)

Kamprath performed a soil incubation study to test soil responses to increasing lime rates in four very acid soils (pH < 5, Al saturation > 50 %). The soils were incubated for 4 weeks at soil moisture capacity and 30°C. The response variables were effective cation exchange capacity (ECEC, sum of exchangeable Al⁺³, Ca⁺², and Mg⁺²), exchangeable Al⁺³, and pH. Lime rates were a multiple of the exchangeable Al⁺³ (meq/100g) in each soil. Thus, lime rates are expressed in meq/100g of soil and not in t/ha or kg/ha.

Kamprath data is available with data(kamp).

Figure 1 shows that Al saturation does not decrease linearly with increasing lime rate. When LR is less than the exch. Al⁺³, 85 to 90% of the lime equivalents react with the exch. Al⁺³. But once the lime rate exceeds the exch. Al⁺³, from a rate of 1 to 1.5 of exch. Al⁺³, only half of that excess reacts with the exch. Al⁺³, and this fraction goes down to 10 to 20% when lime rates are further increased from 1.5 to 2 times the exch Al⁺³.

Therefore, as pH rises, more lime reacts with other Al⁺³ forms (e.g., organic-Al⁺³ complex, interlayer OH-Al⁺³ polymers in vermiculite). Consequently, Al saturation decay with increasing lime rates is not linear but a hyperbola that asymptotically approaches 0. Even when the limed-soil pH is close to 6, and all exch. Al⁺³ should have been precipitated, there is still some exch. Al⁺³ in the soil.

data(kamp)

kamp$Al_sat <- kamp$Al/(kamp$Al + kamp$Ca_Mg) * 100
soils <- unique(kamp$soil)
col <- c('red', 'dark green', 'blue', 'gold')

par(mar = c(4,4,2,2), cex = 0.9)
plot(Al_sat ~ LR, data = kamp, type = "n", 
     xlab = bquote(italic(LR)*"/"*exch.Al^{+3}), 
     ylab = "Al saturation (%)")
for(i in 1:length(soils)){
  si <- kamp[kamp$soil == soils[i],]
  lines(si$LR, si$Al_sat, col = col[i])
}
legend(2.1, 85, legend = soils, col = col, 
       lty = 1, cex = .8, xjust = 1, yjust = 1, bty = "n")  
abline(h = 15, lty = 3)
text(0.5, 18, "15% Al sat")

Figure 1. Al saturation (%) as a function of lime rate in four North Carolina soils.

Kamprath concludes that a lime rate dose (LR, meq/100g) of 1.5 times the exchangeable Al⁺³ is enough for most crops, because it decreases the Al concentration to an Al saturation (Al⁺³ / (Al⁺³ + Ca⁺² + Mg⁺²)) lower than 15% (Figure 1). For very sensitive crops needing less than 15% of Al saturation, the recommended lime rate (meq/100g) is twice the exchangeable Al⁺³.

\[ LR~(meq/100g) = lf \times exch.Al^{+3} \]
Where lf is the lime factor, which equals 1.5 for most crops and 2 for very sensitive crops.

In function lr_ka(), the lf is controlled with the argument lf, whose default value is 1.5.

method = "co": Cochrane et al., (1980)

Cochrane et al. basically takes Kamprath data and develops a formula that estimates the lime rate required to reduce the Al saturation level to the observed Al saturation level in the Kamprath’s trials. Then, the formula is tested in other independent data sets. All soils tested have pH < 5 and Al saturation ≥ 50%.

Cochrane et al. formula is based on the idea that if all the lime reacted with the exch. Al⁺³ and the ECEC does not change, the required lime rate to reach a given Al saturation would be:

\[ LR~(meq/100g) = Al^{+3}_i - Al^{+3}_f = Al^{+3}_i - (TAS/100) \times (Al^{+3}_i + Ca^{+2}_i + Mg^{+2}_i) \]

Where TAS is the target Al saturation (%, divided by 100 to change it to fraction) and cations (Al⁺³, Ca⁺², and Mg⁺²) are the exchangeable cations extracted with a neutral unbuffered salt (e.g., 1N KCl). Subscripts _i and _f refer to the initial and final values.

The sum of the exchangeable cations is the (initial) ECEC of the soil. Only Al, Ca, and Mg are included because these were the cations measured by Kamprath. Other cations can be considered negligible in acid soils. However, if more cations were included in the calculation of ECEC (and Al saturation), it would make sense to include those cations in the formula. Note that this equality is based on the assumption that ECEC does not change with lime rate, so Al_f = TAS × ECEC_i. But ECEC does change.

Since not all the lime reacts with the exch. Al (Figure 1), (and ECEC_f ≠ ECEC_i) the formula is multiplied by a lime factor that equals 1.5 or 2 depending on the relation between initial exch. Al, TAS, and exchangeable cations. Note that in Cochrane et al. paper, and elsewhere, lime factors are also said to be 1.8 and 2.4, but that is for LR expressed in t/ha (considering a 15 cm soil depth and a soil bulk density \(\delta = 1.2 g/cm^3\)).

Cochrane et al. define the following rule:

“the factor 1.5 is replaced by 2 when the estimated liming requirement using the factor 1.5 is greater than the chemical lime equivalent of the exch. Al+3.”

Thus:

\[ LR~(meq/100g) = lf \times [Al_i - (TAS/100) \times (Al^{+3} + Ca^{+2} + Mg^{+2})] \] where

\[\begin{equation} lf=\begin{cases} 1.5, & \text{if $Al^{+3}<1.5 \times (TAS/100) \times (Al^{+3} + Ca^{+2} + Mg^{+2})$}.\\ 2, & \text{otherwise}. \end{cases} \end{equation}\]

This rule is applied by Cochrane et al. to estimate lime requirements for Kamprath data, but using fuzzy boundaries for the lime factor rule (probably to get better accuracy, see Figure 2.).

However, this rule is never included or used in all the Cochrane formula applications and descriptions (see next section). But it is an important rule because Kamprath showed that the higher the lime dose, the lower the proportion that reacts with exch. Al+3 (Figure 1).

As it is presented above, the rule seems quite complicated, but it could be simplified as follows:

\[\begin{equation} lf=\begin{cases} 1.5, & \text{if $TAS > IAS/3$}.\\ 2, & \text{if $TAS \leq IAS/3$}. \end{cases} \end{equation}\]

Where IAS is the inital Aluminum saturation. In other words, if the target Aluminum saturation is more than a third of the initial saturation, the lime factor is 1.5. If the target is equal or less than a third, it is 2. Again, this change in the lime factor (lf) considers that, at small lime doses and high Al saturation, most of the lime reacts with the exch. Al⁺³, but more lime reacts with Al ions released by organic matter –Al complexes or aluminum hydroxides as the pH increases. Following this rule, the difference between Kamprath and Cochrane et al. formula should be small when the TAS is between 10 and 15%.

The data used by Cochrane et al. is available in data(coch). Below I calculate recommended lime rates with limeRate(method = "Cochrane") to compare them to the actual lime rates and the ones calculated by them, as reported in their study.

data(coch)

# initial ECEC
coch$ECEC <- rowSums(coch[, c("exch_Ca", "exch_Mg", "exch_Al", "exch_Ca_Mg")],
                      na.rm = T)

# predicted lime rate by Cochrane 1980 forumla
coch$pLR_meq <- limeRate(coch[, c("exch_Al","ECEC")], method = "Cochrane", 
                         check_Ca = FALSE, TAS = coch$Al_sat)

# convert to kg.ha
coch$pLR_kg.ha <- convert(coch$pLR_meq, 1.2, 20) * 1000

Figure 2 shows the lime rates estimated using Cochrane et al. formula as a function of the actual lime rates for the three different data sets used in their study. Open points represent the estimated lime rate using the formula. The red “x” indicates points that differ from the original publication (i.e., the rule was not followed). Red points are points as published in Cochrane (that do not follow the rule in the formula). Blue points are lime rate estimates not included in the publication because “as lime rate more than double the lime equivalent, formula not applicable.”

par(pty = 's', mfrow = c(1,3), cex.axis = 0.9, tcl = -0.4, mar = c(4,4,2,2),
    mgp = c(2.5,.8,0))

# kamprath 1970
d <- coch[coch$source == "Kamprath1970",]
plot(d$LR_meq, d$pLR_meq,
     xlab = "Actual LR (meq/100g)", 
     ylab = "Estimated LR (meq/100g)") 
mtext("Kamprath (1970)", line = 0.1, cex = 0.9)     
abline(a = 0, b = 1)

# mark differences between results from formula and paper
dd <- d[abs(d$LR_hat/1000 - d$pLR_meq) > 0.1,]
points(dd$LR_meq, dd$pLR_meq, pch = 4, col = "red")
points(dd$LR_meq, dd$LR_hat/1000, pch = 19, col = "red")

# Leon (1978)
d <- coch[coch$source == "Leon1978",]
plot(d$LR_meq, d$pLR_meq,
     xlab = "Actual LR (meq/100g)", 
     ylab = "Estimated LR (meq/100g)") 
mtext("Leon (1978)", line = 0.1, cex = 0.9)     
abline(a = 0, b = 1)

# mark differences between results from formula and paper
dd <- d[abs(d$LR_hat/1000 - d$pLR_meq) > 0.1,]
points(dd$LR_meq, dd$pLR_meq, pch = 4, col = "red")
points(dd$LR_meq, dd$LR_hat/1000, pch = 19, col = "red")

# Brazil field studies
d <- coch[!is.na(coch$LR_kg.ha),]

plot(d$LR_kg.ha, d$pLR_kg.ha, xlim = c(0,8000), ylim = c(0,8000),
     xlab = "Actual LR (kg/ha)", 
     ylab = "Estimated LR (kg/ha)") 
mtext("Brazilian field studies", line = 0.1, cex = 0.9)     
abline(a = 0, b = 1)

# mark differences between results from formula and paper
dd <- d[abs(d$LR_hat - d$pLR_kg.ha) > 100,]
points(dd$LR_kg.ha, dd$pLR_kg.ha, pch = 4, col = "red")
points(dd$LR_kg.ha, dd$LR_hat, pch = 19, col = "red")

# mark in blue points that where omitted in the paper 
dd <- d[is.na(d$LR_hat),]
points(dd$LR_kg.ha, dd$pLR_kg.ha, pch = 19, col = "blue")

Figure 2. Cochrane et al. (1980) lime rate prediction accuracy and inconsistencies.

Cochrane et al. formula works better in very acid soils and low to intermediate lime rates because, in these conditions, most of the lime reacts with the exch. Al+3 (and maybe because ECEC does not change much). It was not tested in moderate acid soils (pH between 5 and 5.5 and Al saturation between 25 and 50%) and for very high lime rates (> 2 times the initial exch. Al). In addition, Cochrane et al. (discretionally) manipulated the rule to change the lime factor between 1.5 and 2 (always getting a better accuracy).

Nevertheless, testing the formula against field studies is tricky because not all the lime might react with the soil in the first year of the experiment, particularly at high lime doses (Figure 3, Brazilian studies). Moreover, it does not make sense to test the formula against very high lime rates because the formula will never predict a ridiculously high dose. Note that “target Al saturation” is the response variable of the experiments and not the independent variable (or treatment).

Thus, despite the inconsistencies and the lack of agreement in field studies and very high lime doses, this widely used formula provides reasonable lime rate recommendations when the target aluminum saturation is not very low, and is one of the best methods available so far.

ACID4: a “modification” of Cochrane et al. formula

Yost et al. (1988) used a “modification” of the Cochrane formula in an expert system developed to make lime recommendations in the humid tropics. The target of the project was Indonesia. They used exchangeable acidity (Al⁺³ + H⁺) instead of exch. Al⁺³ (see the Appendix for an analysis of the difference).

Their formula adaptation is simply a modification of the lime factor (lf) plus a unit conversion. From some (unreachable) data, they calculated that, in their soils, 0.53 cmol of exchangeable acidity is neutralized per cmol of Ca equivalent. Thus, their \(lf = 1/0.53 = 1.9\). They further modify the lf to convert the results from meq/100g to t/ha, assuming a soil bulk density of 1 and an incorporation depth of 15cm. For this reason, they ended with a lf of 1.4. This final formula with \(lf = 1.4\) was used in all ACID4 applications (e.g. Yamoah et al., (1992) and Ngendo (2013, U Gent MS thesis)). Mind the difference between the lf with the unit conversion (\(lf = 1.4\), results in t/ha) and the original lf (= 1.9), which represents the amount of lime needed (meq of Ca/100g soil) to neutralize 1 meq of soil acidity (or exch. Al⁺³) per 100g of soil. Their estimated lf (1.9) is very similar to the one used by Cochrane when the target Al saturation is less than a third of the original Al saturation, which is likely the case for most acid soils and relatively sensitive crops. Note that modifying the lf to account for differences in soil weight (incorporation depth and soil bulk density) is a common practice that adds confusion and results in endless lf values. Also, note that they did not incorporate Cochrane et al. rule to define different lf values depending on the relation between initial and target Al saturation.

method = "nu": NuMaSS (Osmond et al., 2002)

The Integrated Soil Nutrient Management Decision Support System (NuMaSS) was another software developed to provide fertilizer (N and P) and lime recommendations for acid soils with nutrient problems, considering location-specific differences in resources, soil, climate, crop, and management. N, P, and acidity soil constraints were estimated individually, but the final management recommendation was integrated into an economic module that considered costs and benefits.

(Unfortunately, the software is obsolete and unavailable, and that the authors told me that they no longer have the files, program, or data.)

The acidity module considered Al toxicity and/or deficiencies of Ca and Mg, although the main focus was on Al toxicity. The main inputs to estimate Al toxicity were crop critical Al saturation, exchangeable acidity, and ECEC. Default crop critical Al saturation values were available for many crops and varieties. In the absence of exchangeable soil acidity and ECEC information, Al saturation was estimated from pH.

The lime rate was calculated with a modified version of the Cochrane et al. (1980) formula. > “Through our review of lime trials with soils in the tropics, we found that the lime factor differed among soils and between high (>20) and low (<20) % Al saturation values within a soil. The difference in lime factors among soils in the high (>20) % Al saturation range was related to an index of the soil’s clay activity estimated by the cmolc of whole-soil ECEC/l or kg of clay in the soil. The efficiency of CaCO3 in neutralizing exchangeable acidity of soils with low “clay-based ECEC” was lower (higher LF value) than in soils with high “clay-based ECEC” values.”

The formula, as reported by Osmond et al. (2002), is:

\[ LR (t/ha) = lf \times (Al - TAS \times ECEC/100) + 7.5 \times[(19 - TAS) \times ECEC/100] \]

where \[\begin{equation} lf=\begin{cases} 2.5, & \text{if $ECEC/clay < 4.5$}.\\ 1.3, & \text{if $ECEC/clay \geq 4.5$}. \end{cases} \end{equation}\]

Where Al is soil extractable acidity with 1N KCl (exchangeable) and clay is clay fraction in the soil.

Note that when the paragraph quoted above says “the lime factor differed between high and low Al saturation values within a soil,” it is referring to the target Al saturation (TAS), as shown by the second term in the formula: \((7.5 \times [(19-TAS) \times ECEC⁄100]\). This is related to the fact that, for a given soil, more lime is required to decrease one unit of Al saturation as pH rises and Al saturation decreases (as shown in Figure 1). Furthermore, this second term may be only applied when TAS is < 20 (or < 19) to make a two-segment linear response (piecewise) of lime rate to decreasing TAS. Otherwise, it would be a linear response that can take negative values for TAS > 19 (Figure 4).

# different ways in which the formula could be interpreted
# as is
nu_lime1 <- function(tas, exch_al, ecec, clay){
  lf <- ifelse(ecec/clay <= 4.5, 2.5, 1.3) 
  lime <- lf * (exch_al - ecec * tas/100) + 7.5 * ((19 - tas)/100 * ecec)
  return(lime)
  
}

nu_lime2 <- function(tas, exch_al, ecec, clay){
  lf <- ifelse(ecec/clay <= 4.5, 2.5, 1.3) 
  lime <- lf * (exch_al - ecec * tas/100) + pmax(7.5 * ((19 - tas)/100 * ecec), 0)
  return(lime)
  
}

# test functions in two soils with different ECEC/clay activity, for different S
tas <- 50:0

# soil 1: high ECEC/clay, 75% Al saturation
ecec <- 4
clay <- .4

exch_al <- ecec * .75

s1.l1 <- nu_lime1(tas = tas, exch_al = exch_al, ecec = ecec, clay = clay)
s1.l2 <- nu_lime2(tas = tas, exch_al = exch_al, ecec = ecec, clay = clay)

# soil 2: low ECEC/clay, 75% Al saturation
ecec <- 1.5
exch_al <- ecec * .75

s2.l1 <- nu_lime1(tas = tas, exch_al = exch_al, ecec = ecec, clay = clay)
s2.l2 <- nu_lime2(tas = tas, exch_al = exch_al, ecec = ecec, clay = clay)

# plot
par(mfrow = c(1,2), mgp = c(2, .8, 0), mar = c(3,3,2,1), cex = 0.9)
plot(tas, s1.l1, type = "l", col = "red", xlab = "TAS (%)", ylab = "LR (t/ha)", 
    xlim = c(50,0), main = "Formula as is")
lines(tas, s2.l1, col = "blue")
abline(h = 0, lty = 3)
legend(50, 10, legend = c("High ECEC", "Low ECEC"), lty = 1, 
      cex = .8, col = c("red", "blue"), xjust = 0, yjust = 1, bty = "n")  
abline(h = 0, lty = 3)

plot(tas, s1.l2, type = "l", col = "red", xlab = "TAS(%)", ylab = "",
    xlim = c(50,0), main = ("Piecewise"))
lines(tas, s2.l2, col = "blue")
legend(50, 10, legend = c("High ECEC", "Low ECEC"), lty = 1, 
      cex = .8, col = c("red", "blue"), xjust = 0, yjust = 1, bty = "n")  
abline(h = 0, lty = 3)

Figure 3. Lime rate (LR, t/ha) as predicted by two interpretations of NuMaSS formula for two soils with different ECEC/clay.

The formula predicts lime rates in t/ha by assuming an incorporation depth of 15 cm and a soil bulk density of 1 g/cm³. Incorporation depth is stated clearly. The formula description says that exch. acidity and ECEC can be expressed in cmol/l or kg, and that the program converts exchangeable acidity and ECEC values from weight-by-weight lab units to weight-by-volume at the field level. Thus, if soil bulk density is different from 1g/cm3, the ECEC, and exch. acidity values are transformed from cmol/kg to cmol/l by multiplying those values with the soil bulk density (kg/l). Converting exchangeable acidity and ECEC values is the same as converting the lime factor (as done by Yost et al. (1988) and others) and also equivalent to calculating the lime rate in meq/100g and then transforming the result. Thus, for compatibility with other formulas, the formula was rewritten converting the lime factors so that the result is in meq/100g soil. Taking this into account and the fact that the second term cannot result in negative values, the formula implemented in lr_nu() is:

\[\begin{equation} LR (meq⁄100g) = \begin{cases} lf \times [Al - (TAS/100 \times ECEC)] + 10 \times [(19-TAS)/100 \times ECEC], & \text{if $TAS < 19$}\\ lf \times [Al - (TAS/100 \times ECEC)] , & \text{ if $TAS \geq 19$} \end{cases} \\ \text{where:} \\ lf = \begin{cases} 3.33, & \text{ if $ECEC/clay < 4.5$} \\ 1.73, & \text{ if $ECEC/clay \geq 4.5$} \end{cases} \end{equation}\]

The NuMaSS formula seems to offer a good theoretical improvement against the Cochrane formula because it considers that Al saturation response to increasing lime rates is not linear (Figure 1) and that soils differing in their ECEC activity have different responses.
However, Figure 5 shows that NuMaSS does a worse job predicting incubation studies results, particularly when TAS is low (< 10). The rule for TAS < 19 can severely over-predict lime rates and should be revised.

# Formulas evaluation ######################
coch$LR <- dplyr::coalesce(
  coch$LR_meq,
  coch$LR_kg.ha/1000
)
coch$pLR_meq <- NULL
coch$pLR_kg.ha <- NULL

coch$nu_LR <- limeRate(coch[, c("exch_Al","ECEC", "clay")], method = "numass", TAS = coch$Al_sat, check_Ca = FALSE)

coch$nu_LR <- ifelse(is.na(coch$LR_kg.ha), 
                     coch$nu_LR, 
                     convert(coch$nu_LR, 1.2, 15))

coch$LR <- ifelse(is.na(coch$LR_meq), coch$LR_kg.ha/1000, coch$LR_meq)

nams <- unique(coch$source)
n <- length(nams)
cols <- viridis::viridis(41)

par(mfrow = c(1,2), mar = c(4,4,2,2), mgp = c(2, .8, 0), cex = 0.9)
plot(coch$LR, coch$co_LR, type = "n",
     xlab = "Actual lime rate", 
     ylab = "Estimated lime rate")
mtext("Cochrane formula")

for(i in 1:n){
  d <- coch[coch$source == nams[i],]
  points(d$LR, d$co_LR, bg = cols[pmin(d$Al_sat, 40)+1], 
         pch = ifelse(is.na(d$LR_kg.ha), 21, 22))
}

abline(a = 0, b = 1)

plot(coch$LR, coch$nu_LR, type = "n", ylim = c(0,20), xlim = c(0,20),
     xlab = "Actual lime rate", 
     ylab = "Estimated lime rate")
mtext('NuMaSS formula')
for(i in 1:n){
  d <- coch[coch$source == nams[i],]
  points(d$LR, d$nu_LR, bg = cols[pmin(d$Al_sat, 40)+1], 
         pch = ifelse(is.na(d$LR_kg.ha), 21, 22))
}
abline(a = 0, b = 1)

Figure 4. Estimated vs. actual lime rates for Cochrane and NuMaSS formulas. The colors of the point indicate the target Al saturation (high TAS = yellow, low TAS = blue). Circles are observations obatined in incubation studies and squares in field studies.

It should be noted that this way of testing the formulas is far from ideal. The target Al saturation is an independent variable in the formulas, but the response variables of the experiment. On the contrary, lime rate is the independent variable in experiments but a dependent variable in formulas. Thus, there might be very high experimental lime rates that a formula cannot predict.

method = "my" My lime recommendation formula

TBD. . . . .

Base Saturation formula

method = "bv Brazil V mehtod

This method has been used in the state of Sao Paulo, Brazil, at least for the last thirty years. It is based on the “base saturation”, which is the sum of all exchangeable bases divided the Cation Exchange Capacity at pH 7 (CEC_7). The sum of all exchangeable cations (bases plus acidity (Al and H)) is the Effective Cation Exchange Capacity (ECEC), which is always lower than the CEC_7 in acid soils, simply because at higher pH, there are more pH dependent negative charges available. The difference between ECEC and CEC 7 is related to the buffer capacity of the soil. Thus, the Brazil V method somehow takes this buffer capacity into account.

On the other hand, the method considers that all the applied lime increases the base saturation of the soil, thus, liming has a 100% efficiency in increasing base content. The formula estimates how much lime is required to increase the base saturation of the soil to the required level for the crop. It is considered that most crops required base saturation levels between 40 and 70%. The limeRate function provides some default values for different crop types.

The Brazil V formula is

\[ LR = CEC_{pH 7} \times (V_f - V_i) \] where \(V_f\) is the final base saturation and \(V_i\) is the initial base saturation. And base saturation equals: \(V = (Exch.Ca + Exch. Mg + Exch. K + Exch. Na)/CEC_{pH7}\)
For convenience, in limeRate the formula is implemented as:

\[ LR = CEC_7 \times V_f - (ECEC - exch. acidity) \]

Note that \(CEC_7 \times V_f\) equals the final sum of exchangeable bases, while \(ECEC - exch. acidity\) equals the initial sum of exhangeable bases.

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Appendix: exchangeable Al⁺³ vs. exchangeable acidity.

Kamprath (1970) and most old papers (> 40 y) only reported exch. Al⁺³ values, and they never mentioned exchangeable acidity. Kamprath extracted exch. Al⁺³ with 1N KCl, and then it was measured by titration. They did not specify the titration method used (potentiometric or conductimetric). If titration is performed with a base to neutrality (potentiometric), then exchangeable acidity is measured (Kamprath, 1984; Lin & Coleman, 1960). But one may assume that exchangeable acidity is almost the same as exch. Al⁺³ when soil pH is between 4 and 5.5, because “exchangeable H⁺ is normally present in measurable quantities only at pH values less than 4” while exch. Al⁺³ is significant at soil pH values less than 5.5 (Bohn et al., 2002).

Moreover, distinguishing between acidity components (H⁺, Al⁺³, Fe⁺²) is not easy. It can be done by a conductometric titration (Bohn et al., 2002) or by taking extra steps using KF and HCl on the alreadt titrated KCl-extracted soil solution (Sims, 1996). Note that in either case, all acidity components are quantified (Al⁺³ + H⁺), so it can be expected that all are reported.

Therefore, most likely all exch. Al⁺³ reported in the old literature (without mentioning H⁺) are measures of exchangeable acidity (but primarily caused by Al) as it is very difficult to discern between acidity components (Bohn et al., 2002). In his PhD thesis, Jose Salinas states this clearly:

“Exchangeable acidity was determined by titration of the filtrate with 0.01N NaOH using bromthymol blue as an indicator. Exchangeable acidity was considered to be due to exchangeable Al (Lin and Coleman, 1960)” (pp 79)

For this reason, in the limer package, all lime recommendation formulas based on Aluminum saturation require exchangeable acidity instead of exchangeable aluminum.

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