How accurate are pedotransfer functions for bulk density for Brazilian soils?

ABSTRACT:

The aim of this study was to evaluate the performance of pedotransfer functions (PTFs) available in the literature to estimate soil bulk density (ρb) in different regions of Brazil, using different metrics. The predictive capacity of 25 PTFs was evaluated using the mean absolute error (MAE), mean error (ME), root mean squared error (RMSE), coefficient of determination (R²) and the regression error characteristic (REC) curve. The models performed differently when comparing observed and estimated ρb values. In general, the PTFs showed a performance close to the mean value of the bulk density data, considered as the simplest possible estimation of an attribute and used as a parameter to compare the performance of existing models (null model). The models developed by Benites et al. (2007) (BEN-C) and by Manrique and Jones (1991) (M&J-B) presented the best results. The separation of data into two layers according to depth (0-10 cm and 10-30 cm) demonstrated better performance in the 10-30 cm layer. The REC curve allowed for a simple and visual evaluation of the PTFs.

Keywords:
REC curve; model evaluation; soil database; tropical soil

Introduction

Bulk density (ρb) is an important soil physical property known to affect soil water movement, root growth, seed germination and root density (Mouazen et al., 2003; Dexter, 2004). This property has received particular attention due to its importance in weight-to-volume conversions used to assess soil organic carbon (OC) stocks. Soil is known to contain the largest terrestrial carbon pool and can act as an important sink or source for atmospheric CO₂.

Despite their importance, ρb databases are in short supply because direct measurements of undisturbed samples are labor-intensive and time-consuming. Furthermore, ρb is highly variable in space and time (Alletto and Coquet, 2009).

Pedotransfer functions (PTFs) (Bouma, 1989) have been widely used for estimating ρb using soil properties which are easier to measure and are available in most databases. These PTFs have been developed from specific datasets using OC and texture data as input parameters (Curtis and Post, 1964; Alexander, 1980; Federer, 1983; Grigal et al., 1989; Huntington et al., 1989; Manrique and Jones, 1991; Bernoux et al., 1998; Tomasella and Hodnett, 1998; Kaur et al., 2002; Prévost, 2004; De Vos et al., 2005; Périé and Ouimet, 2008; Han et al., 2012; Al-Qinna and Jaber, 2013; Hong et al., 2013; Nanko et al., 2014). Benites et al. (2007) also used the sum of basic cations (SB) as an input parameter.

A number of studies have also tested the performance of available PTFs and observed that these functions are relatively inaccurate when applied to different environments (Kaur et al., 2002; De Vos et al., 2005; Benites et al., 2007; Al-Qinna and Jaber, 2013; Nanko et al., 2014). These studies assessed the performance of PTFs based on several error measures required for a reliable evaluation. The root mean squared error (RMSE) and mean error (ME) are the most commonly used measures (De Vos et al., 2005; Benites et al., 2007; Al-Qinna and Jaber, 2013; Nanko et al., 2014).

The aim of this study was to evaluate the predictive capability of PTFs available in the literature to estimate soil ρb in different regions of Brazil, using different metrics (mean absolute error (MAE), root mean squared error (RMSE), mean error (ME) and regression error characteristic (REC) curve).

Materials and Methods

Selected pedotransfer functions (PTFs)

We evaluated 25 PTFs available in the literature. The PTFs selected, R² values and the size of the data set used to generate the PTFs are shown in Table 1. PTF equations were adjusted to the same units for better comparison. The details about land use, location, ρb range, and the method used to determine the ρb, which was extracted from the original papers, are presented in Table 2.

Table 1
Published pedotransfer to estimate bulk density. Sample size (n) and R² were taken from the original paper.

Reference	PTF	R2	n	Code
Alexander (1980)	ρb = 1.66 – 0.308 (OC)1/2	0.46	721	ALEX
Benites et al. (2007)	ρb = 1.56 – 0.0005 (Clay*10) – 0.01 (OC*10) – 0.0075 SB	0.66	1542	BEN-A
Benites et al. (2007)	ρb = 1.5688 – 0.0005 (Clay*10) – 0.009 (OC*10)	0.42	1542	BEN-B
Benites et al. (2007)	ρb = 1.5224 – 0.0005 (Clay*10)	0.63	1542	BEN-C
Bernoux et al. (1998)	ρb = 1.398 – 0.0047 Clay – 0.042 OC	0.49	323	BERX
Curtis and Post (1964)	Log10(ρb*100) = 2.09963-0.00064log10(LOI) – 0.22302 log10(LOI)2	0.96	~144	C&P
Federer (1983)	ln(ρb) = −2.31 – 1.079 ln[(OM/100)] – 0.113 ln[(OM/100)]2	NI	NI	FED
Grigal et al. (1989)	ρb = 0.075 + 1.301exp (-0.06 LOI)	0.95	800	GRIG
Harrison and Bocock (1981)	ρb =1.558 – 0.728 log10(LOI)	0.81	539	H&B-A
Harrison and Bocock (1981)	ρb =1.729 – 0.769 log10(LOI)	0.58	538	H&B-B
Han et al. (2012)	ln(ρb) = 0.5379 – 0.0653[(10 OM)]0.5	0.67	1560	HAN
Hong et al. (2013)	ρb =1.02 – 0.156 ln(OM)	0.45	642	HONG
Huntington et al. (1989)	ln ρb = −2.39 – 1.316 ln [(OM/100)] – 0.167 (ln [(OM/100)]2	0.75	60	HUNT-A
Huntington et al. (1989)	ln ρb = 0.263 – 0.147 ln OC – 0.103 (ln OC)2	0.72	60	HUNT-B
Jeffrey (1970)	ρb =1.482 – 0.6786 log10(LOI)	0.82	80	JEF
Kaur et al. (2002)	ln ρb = 0.313 – 0.191 OC + 0.02102 Clay – 0.000476 (Clay)2 – 0.00432 Silt	0.62	224	KAUR
Leonavičiute (2000)	ρb =1.70398 – 0.00313 Silt + 0.00261 Clay – 0.11245 OC	NI	135	LEO-A
Leonavičiute (2000)	ρb = 1.07256 + 0.032732 ln(Silt) + 0.038753ln(Clay) + 0.078886 ln(Sand) – 0.054309 ln(OC)	NI	858	LEO-B
Manrique and Jones (1991)	ρb = 1.66 – 0.318 (OC)1/2	0.41	19651	M&J-A
Manrique and Jones (1991)	ρb = 1.51 – 0.113 OC	0.36	19651	M&J-B
Perie and Ouim (2008)	ρb = 1.977 + 4.105(OM/100) – 1.229ln[(OM/100)] – 0.103 ln[(OM/100)]2	0.82	125	P&O
Prévost (2004)	ln(ρb) = −1.81 – 0.892*ln[(OM/100)] – 0.092 ln[(OM/100)]2	0.77	414	PREV
Ruehlmann and Körschens (2009)	ρb = (2.684 - (140.943b)) exp(-bOC)	0.98	609	R&K
Tomasella and Hodnett (1998)	ρb = 1.578 – 0.054 OC – 0.006 Silt – 0.004 Clay	0.59	613	T&H
Tamminen and Starr (1994)	ρb = 1.565 – 0.2298(LOI)0.5	0.61	158	T&S

• ρb = bulk density (g cm^-3); OM = organic matter (%); OC = organic carbon (%); SB = sum of basic cations (cmol_c kg⁻¹); LOI = loss-on-ignition (%); NI = not reported in the original paper; b = coefficient for soil groups proposed by the authors (b = 0.006).

Table 2
Land use, location, values of bulk density and method to determine bulk density for the data set used in the development of the tested pedotransfer functions (PTFs). This information was taken from the original papers of the tested pedotransfer functions.

PTF	Land use	Location	Bulk Density (g cm-3)	Method
ALEX	Alluvial soils and uplands	California - EUA	0.45-1.92/0.24-1.98	clod/core
BEN	NR	Brazil	0.13-2.25 (1.36)	core
BERX	NR	Amazon - Brazil	0.74-1.58 (1.18)	clod
C&P	Forest	Vermont - EUA	NR	core
FED	Forest	New England	NR	core
GRIG	Florest	North Central USA	0.35-1.90	core
H&B	Various	Various	0.03-1.87	various
HAN	Forest	China	0.12-1.82 (1.32)	NR
HONG	NR	Andisols - Global Dataset	NR	NR
HUNT	Spodosols under forest	New Hampshire - EUA	NR	Pits
JEF	Various	Australia and England	0.05-1.8	core
KAUR	Various	Uttaranchal - Índia	0.85-1.79 (1.36)	core
LEO	NR	Lithuania	0.44-2.14 (1.68)	core
M&J	Various	EUA	1.2-1.5	NR
P&O	Forest	Quebec - Canada	NR	core
PREV	Forest	Quebec - Canada	0.10-1.96	core
R&K	Various	Various	0.03-2	core
T&H	NR	Amazon - Brazil	0.95-1.6	core
T&S	Forest	Finland	0.48-1.84	core
Data set	Forest, Pasture, ICL, ICLF	Brazil	0.81-1.84 (1.42)	core

• NR = not reported in the original paper; ICL = integrated crop-livestock system; ICLF = integrated crop-livestock-forest system.

Data set

The original data set used to evaluate the performance of the 25 PTFs consisted of 222 soil profiles (888 soil layers) distributed in different biomes in Brazil and with different uses (native vegetation, pasture, integrated crop-livestock and integrated crop-livestock-forest systems) (Figure 1).

Figure 1
Map with the location of points sampled.

Disturbed and undisturbed soil samples were collected from four depths (0-5, 5-10, 10-20 and 20-30 cm) for chemical and physical analysis (Embrapa, 1997). The particle size distribution of sand (2.00-0.05 mm), silt (0.05-0.002 mm), and clay (< 0.002 mm) was determined by the hydrometer method. Soil organic matter (OM) was determined using the colorimetric method. The OC content was determined by using the elemental analyzer in Piracicaba, São Paulo, Brazil. Bulk density was obtained for each layer using the core method, by means of cylinders with volumetric rings 0.053 m high and 0.05 m in diameter. Detailed information can be found in Assad et al. (2013).

We eliminated samples that had a missing ρb value or whose sum of particle size fractions (sand, silt, and clay) did not equal 100 %. The final data set consisted of 884 soil samples. The evaluation of the 25 PTFs was first performed by considering the whole data set (884 samples). In a second analysis, the data set was divided into two subgroups according to soil depth: the layer between 0 and 10 cm (0-10 cm) and the layer between 10 and 30 cm (10-30 cm).

Evaluation criteria

The most useful approach to an evaluation of predictive models should be based on a set of complementary indices (Donatelli et al., 2004; Nanko et al., 2014). In the present study, the predictive capacity of the 25 PTFs was evaluated using the mean absolute error (MAE), RMSE, ME and R². MAE is the value of the error in absolute terms, and lower values indicate better predictive ability. RMSE is the standard deviation of the error in the prediction, and, again, lower values indicate better model performance. ME quantifies systematic errors and indicates tendencies to overestimate or underestimate. R² represents the fraction of the total variance that is explained by the model. Higher R² values are desirable.

Regression error characteristic curve - REC curve

The evaluation of the PTFs was complemented by the use of the REC curve. The regression error characteristic (REC) curve is a promising alternative for the evaluation of PTFs. The REC curve allows for comparisons of the performance of different models simultaneously using visual assessment, which facilitates interpretation by the user (Mittas and Angelis, 2010).

The REC curve represents the cumulative distribution function of the errors. On the x-axis the REC curve plots the error and on the y-axis the accuracy of a regression function (Mittas and Angelis, 2010). Accuracy is defined as the fraction of points that fit within an error level, ranging from 0 to 1. Using this construction, we can evaluate the level of accuracy as the number of points for a given value of error.

REC curves are built in a way that makes inspecting model performance a simple task of checking which model is closest to the (0, 1) point (upper left corner) (Figure 2A). If the REC curve of one model is above the others, it is an indication that it is the best model for that metric, be it absolute error or squared error, or their relative forms (e.g., model c in Figure 2B). Thus, when comparing models a and b, if the REC curve from model a overlies the curve of model b, we can affirm that model a is superior to model b (Figure 2B). Model c in Figure 2B is superior to the others.

Figure 2
A) Illustration of the Area Over Curve (AOC) of a model; B) Comparison of three models by their REC (regression error characteristic) curve; model c representing near perfect behavior, the model with the highest performance, which in turn is better than model b.

The REC curve also allows us to estimate the model error by measuring the area over the curve (AOC). If the selected error measure is the squared error, the AOC corresponds to an underestimation of the mean squared error (MSE), and for the absolute error, the AOC corresponds to an underestimation of the MAE.

Finally, the REC curve allows for a comparison of the quality of a particular model with the simplest possible estimation of an attribute from a population. In this study, the mean prediction corresponds to the simplest possible estimation of an attribute, and we used these values as a parameter to compare the performance of existing models. If the use of the mean values is better than a particular model, this model is not recommended for that situation. The mean value will hereafter be referred to as the null model.

When used in such manner, comparing models with REC curves is no different from using the MAE or RMSE. The benefits of the REC curve are related to other easily visualized statistics, which can be the R² or the Kolmogorov-Smirnov Statistic to check if two samples are generated from the same distribution. Furthermore, the REC allows for quick visualization of the error median (50 % horizontal line) or any level of confidence for different readers, e.g. one can look at the errors for 80 % confidence or 95 %.

REC curves were constructed following the steps proposed by Mittas and Angelis (2010): 1 - The model was set up, and the error in the predictions was evaluated; 2 - The error values were sorted in ascending order; 3 - For each error value that was not repeated, a point with the error value and the percentage of points with smaller errors were included in the graph. Step 1 was not necessary because we used available PTFs (Table 1). The RMSE was used as a measurement of error. Comparisons between curves were also made, by applying the Wilcoxon signed rank test at the 0.05 level, with the Holm correction for multiple comparisons. This test assesses whether population mean ranks differ (paired difference test).

Results and Discussion

General evaluation

The descriptive statistics of the data set are shown in Table 3. The data set used has a wide range of ρb values, with a minimum of 0.81 g cm^-3, a maximum of 1.84 g cm^-3, and a mean value of 1.42 g cm^-3 (Table 3). The PTFs tested were developed from data sets with different uses, origins, ρb ranges and methods of determination (Table 2). The range of ρb values in the data set used for the comparisons was within the range of values observed in the data sets used to generate the PTFs ALEX, BEN, GRIG, HAN, H&B, JEF, KAUR, LEO, PREV, R&K and T&S (Table 2 and Table 3). The PTFs BERX and HUNT used methods different from those used in this study; the PTFs ALEX and H&B used more than one method; and the PTFs HAN, HONG, and M&J did not inform the method used. The others used the core method.

Table 3
Descriptive statistics of the data set used to evaluate the 25 pedotransfer functions.

Statistics	Silt	Clay	Sand	OC	OM	ρb	SB
	-----------------------------------------%---------------------------------------					g cm–3	cmolc kg–1
All data set (n = 884)
Min-Max	1 – 67	0.1 – 76	2 – 95	0.1 – 8.3	0.2 – 13.2	0.81 – 1.84	0.3 – 30.5
Mean (Std dev)	17	28	55	1.4	2.5	1.42	4
Std dev	12	17	24	0.9	1.4	0.2	3.9
1st – 3rd quartile	9 – 22	15 – 40	38 – 73	0.7 – 1.8	1.5 – 3.1	1.26 – 1.56	1.6 – 5.1
Median	13	25	60	1.1	2.1	1.42	2.9
Topsoil (n = 442)
Min-Max	1 – 64	0.1 – 76	3 – 95	0.1 – 8.3	0.6 – 13.2	0.81 – 1.8	0.3 – 30.4
Mean (Std dev)	17	27	56	1.7	2.9	1.39	4.9
Std dev	12	17	24	1.1	1.6	0.2	4.1
1st – 3rd quartile	9 – 22	14 – 37	39 – 74	0.8 – 2.2	1.9 – 3.7	1.25 – 1.54	2.3 – 6.4
Median	13	24	62	1.4	2.7	1.4	3.7
Subsoil (n = 442)
Min-Max	1 – 67	0.2 – 76	2 – 95	0.1 – 4.6	0.2 – 8.6	0.88 – 1.84	0.3 – 30.5
Mean (Std dev)	17	30	54	1.1	1.9	1.41	3.2
Std dev	12	18	24	0.7	1.1	0.2	3.6
1st – 3rd quartile	9 – 22	16 – 42	35 – 72	0.6 – 1.5	1.2 – 2.4	1.26 – 1.57	1.2 – 4
Median	13	26	58	0.9	1.7	1.44	2.2

• Min = minimum; Max = maximum; Std dev = standard deviation; OC = organic carbon; OM = organic matter; ρb = bulk density; SB = sum of basic cations.

The models performed differently when observed and estimated ρb values were compared (Figure 3 and Table 4). The ρb values were underestimated, except for the R&K, H&B-B and LEO (A and B) functions (Figure 3, Table 4). Underestimations in the prediction of ρb were also observed by De Vos et al. (2005), which they attributed to the high proportion of topsoil data used in the calibration of the PTFs. These authors used forest soil data, in which the surface ρb tended to be lower than the subsurface values (Tamminen and Starr, 1994; De Vos et al., 2005), which can lead to underestimated ρb values. The data set used in this study are composed of soil data from the upper 30 cm of the soil profile, considered as topsoil (De Vos et al., 2005; Martin et al., 2009). Additionally, 40 % of our data is from natural vegetation (ρb = 1.28 g cm^-3) and integrated systems (crop-livestock ρb = 1.33 g cm^-3; crop-livestock-forest ρb = 1.39 g cm^-3), which presented a lower topsoil ρb than pasture (ρb = 1.45 g cm^-3). Data only from topsoil and under different uses may explain the underestimates.

Table 4
Mean absolute error (MAE), root mean squared error (RMSE), mean error (ME), coefficient of determination (R²), and area over the REC curve (AOC) with mean squared error (sqr) and with absolute error (abs) for published pedotransfer functions (PTFs) evaluated using bulk density data for all samples. The Wilcoxon Test was performed for differences between distributions of absolute errors.

All Data set
PTFs	MAE	RMSE	ME	R2	AOC_sqr	AOC_abs	Wilcoxon Test
HUNT-A	0.27	0.31	-0.24	0.05	0.10	0.27	hi
HUNT-B	0.20	0.24	-0.17	0.34	0.06	0.20	bcd
ALEX	0.15	0.19	-0.08	0.32	0.03	0.15	a
M&J-A	0.16	0.19	-0.10	0.32	0.04	0.16	e
M&J-B	0.14	0.17	-0.05	0.32	0.03	0.14	f
T&H	0.16	0.19	-0.11	0.40	0.04	0.16	ae
BERX	0.22	0.26	-0.19	0.31	0.07	0.22	g
KAUR	0.31	0.38	-0.29	0.41	0.15	0.31	hi
BEN-A	0.19	0.23	-0.15	0.35	0.05	0.19	bcd
BEN-B	0.16	0.19	-0.10	0.35	0.04	0.16	e
BEN-C	0.15	0.18	-0.02	0.19	0.03	0.15	f
LEO-A	0.19	0.24	0.17	0.27	0.06	0.19	bcdkm
LEO-B	0.20	0.25	0.17	0.32	0.06	0.20	bcdgkm
R&K	0.42	0.47	0.42	0.31	0.22	0.42	n
FED	0.27	0.31	-0.25	0.16	0.10	0.27	i
T&S	0.22	0.26	-0.18	0.14	0.07	0.22	gm
HAN	0.20	0.24	-0.14	0.14	0.06	0.20	b
JEF	0.21	0.25	-0.14	0.12	0.06	0.21	k
H&B-A	0.19	0.23	-0.08	0.12	0.05	0.19	c
H&B-B	0.19	0.24	0.08	0.12	0.06	0.19	bcdek
HONG	0.50	0.53	-0.50	0.12	0.28	0.50	l
GRIG	0.23	0.23	-0.20	0.27	0.07	0.23	j
C&P	0.26	0.30	-0.24	0.17	0.09	0.26	h
PREV	0.19	0.23	-0.13	0.16	0.05	0.19	d
P&O	0.19	0.24	-0.10	0.14	0.06	0.19	bd
Null	0.17	0.20	0.00	NA	0.04	0.17	ae

• NA = not available; Null = null model; models followed by the same letter did not differ in the Willcoxon Signed Rank test at the 0.05 level with the Holm correction for multiple testing.

Figure 3
Values observed (OBS) versus estimated (EST) of 25 PTFs (pedotransfer functions) evaluated in a data set of soils from different regions of Brazil with bulk density data for depths up to 30 cm. The dashed line corresponds to 1:1.

Lower ME values were observed for BEN-C and M&J-B indicating that these models were less biased (Table 4). The most biased models were HONG, R&K, and KAUR. Out of the 12 models evaluated by De Vos et al. (2005), KAUR was the most biased and showed a high ME value. In the present study, this model had the third highest ME value (Table 4). The data used by De Vos et al. (2005) presented a ρb mean value (1.44 g cm^-3) very similar to the data used here (1.42 g cm^-3). However, the range of ρb (0.22 to 1.96 g cm^-3) was wider than what we recorded (0.81 to 1.84 g cm^-3). On the other hand, the most biased models tested by Nanko et al. (2014) were H&B and ALEX, which were among the models with lower values. They used data from volcanic soils, with a range of ρb between 0.13 and 1.78 g m^-3 and a very low mean of 0.6 g cm^-3.

The best indices (MAE, RMSE, and ME) were observed in the M&J-B, ALEX, BEN-B, BEN-C, M&J-A and T&H models. Out of the models with lower errors (MAE, RMSE, and ME), the T&H model had the highest R² value and, therefore, represented a more realistic model. When considering the models developed from the data from Brazilian soils (BEN-A, BEN-B, BEN-C, BERX, and T&H), we observed that three of them are among the best models: BEN-B, BEN-C and T&H.

The worst indices were observed in the HONG, R&K, and KAUR models. Furthermore, HONG, R&K, BEN-C, LEO-B, and C&P models provided similar estimates of ρb (Figure 3) for a data set with a broad range of values (0.86 g cm^-3 to 1.8 g cm^-3) (Table 3). This finding reveals that these models do not provide reliable predictions of ρb. The performance of the PTFs using data sets that differ from those used in their development is uncertain and may not be satisfactory, especially when the data sets are sourced from different geographic areas (Wösten et al., 2001) (Table 5).

Table 5
RMSE (root mean squared error) (First value) and R² (Second value) for the PTFs (pedotransfer functions) Alexander et al. (1980) (ALEX), Manrique and Jones (1991) (M&J) and Tomasella and Hodnett (1998) (T&H) and Kaur et al. (2002) (KAUR), when applied to different data sets.

References	RMSE/R2					Soil use and Location
	ALEX	M&J	T&H	R&K	KAUR
Nanko et al. (2014)	0.32/0.64	0.30/0.64	-	-	-	Forest soils affected by volcanic ash (Japan)
Al-Qinna and Jaber (2013)	0.33/0.22	0.34/0.23	0.42/0.258	0.14/0.21	0.52/0.21	Arid environment in North Jordan
De Vos et al. (2005)	0.30/0.58	0.32/0.58	-	-	0.56/0.58	Forest in Belgium
Kaur et al. (2002)	0.19/-	0.20/-	0.18/-	-	-	Four micro-watersheds in Almora, India
Han et al. (2012)	0.14/0.59	0.35/0.46	0.21/0.22	-	0.26/0.45	Forest in China
Benites et al. (2007)	-	0.27/-	0.24/-	-	-	Different soil use in Brazil

In addition to the difficulty associated with extrapolating these models due to the specificity of the data sets used in their development (Table 3), the method used to generate the PTFs also appears to affect the results (Martin et al., 2009; Jalabert et al., 2010; Suuster et al., 2011). These methods are quite varied, ranging from simple regressions to more powerful methods such as neural networks, regression trees (Martin et al., 2009; Jalabert et al., 2010; Ghehi et al., 2012), and the nearestneighbor method (Nemes et al., 2010). In the case of the PTFs assessed in our study, the methods used were mostly simple or multiple regressions using the least squares method.

The analysis of input variables did not present any concluding remark. The T&H model, using OC and two granulometry fractions was one of the best five models (M&J-B, ALEX, BEN-B, BEN-C, M&J-A, and T&H); despite one of the worst models (KAUR) also using OC and two granulometry fractions. The other models used only OC or OM and one granulometry fraction. Therefore, when considering the origin of the data set used to develop the model, the number, and type of input variables, the model generated from a Brazilian data set with OC and two granulometry fractions as input was able to predict ρb more effectively.

When comparing the observed and predicted values, certain patterns in results appear inadequate to act as a good predictor (Figure 3). An exhaustive discussion of these patterns is beyond the scope of this study; thus, two patterns were observed: the horizontal lines (e.g., BEN-C) and the ‘flat top’ (e.g., HUNT-A). The horizontal line found in BEN-C is easily explained by the small magnitude of the angular coefficients (10^-3-10^-5) compared with the linear coefficient (1.5688). Although the data set used in this study contains ρb values between 0.81 and 1.84 g cm^-3, the BEN-C model predicted values between 1.14 and 1.51 g cm^-3. This range is even narrower for the LEO-B model, which predicted values between 1.33 and 1.71 g cm^-3. The previous analysis developed for BEN-C and LEO-B can be extended to the R&K and C&P models.

The ‘flat top’ pattern was observed in the HUNT-A, HUNT-B, and KAUR models and, to a lesser extent, in LEO-A and FED. An evaluation of the parameters of the HUNT-A equation and the values of the data set allowed us to conclude that the predicted values would lie between 1.04 and 1.36 g cm^-3. The maximum value is lower than the average ρb value for the data set (1.42 g cm^-3), resulting in a flat aspect on the graph.

When the data set was separated according to depth (0-10 cm and 10-30 cm), the values of the indices showed only slight variations (Table 6). The M&J-B, BEN-B, BEN-C, ALEX, M&J-A, and T&H continued to perform well, and the HONG, R&K, and KAUR models had the worst performance. The lowest RMSE values were observed for the BEN-C and M&J-B and the highest for the HONG model for both soil depths.

Table 6
Mean absolute error (MAE), root mean squared error (RMSE), mean error (ME), coefficient of determination (R²) and area over de REC curve (AOC) with absolute error for published pedotransfer functions (PTFs) evaluated using bulk density data for two classes of depth (0 to 10 cm and 10 to 30 cm). The Wilcoxon Test (WT) was performed for differences between distributions of absolute errors.

PTFs	Depth of 0 to 10 cm						Depth of 10 to 30 cm
	MAE	RMSE	ME	R2	AOC	WT	MAE	RMSE	ME	R2	AOC	WT
HUNT-A	0.26	0.30	-0.23	0.12	0.26	a	0.28	0.32	-0.24	0.01	0.28	a
HUNT-B	0.22	0.26	-0.20	0.34	0.22	bcd	0.18	0.21	-0.14	0.41	0.18	bcdefghi
ALEX	0.16	0.20	-0.11	0.33	0.16	ef	0.15	0.18	-0.06	0.36	0.15	jk
M&J-A	0.17	0.20	-0.12	0.33	0.17	ghij	0.15	0.18	-0.07	0.36	0.15	lm
M&J-B	0.14	0.18	-0.07	0.34	0.14	kl	0.14	0.17	-0.03	0.39	0.14	jl
T&H	0.16	0.19	-0.11	0.38	0.16	egh	0.16	0.19	-0.11	0.41	0.16	kmno
BERX	0.21	0.25	-0.19	0	0.21	bcd	0.23	0.26	-0.20	0.30	0.23	pqr
KAUR	0.34	0.41	-0.32	0	0.34	mn	0.29	0.36	-0.26	0.42	0.29	apst
BEN-A	0.20	0.24	-0.17	0.35	0.20	opq	0.18	0.21	-0.13	0.36	0.18	bcdefghiuvwx
BEN-B	0.17	0.20	-0.11	0.35	0.17	fij	0.16	0.19	-0.09	0.35	0.16	n
BEN-C	0.14	0.18	0	0.18	0.14	k	0.15	0.18	-0.04	0.22	0.15	jl
LEO-A	0.18	0.22	0.15	0.30	0.18	bcefghijopr	0.21	0.26	0.19	0.28	0.21	bcegqyzA
LEO-B	0.20	0.25	0.18	0.29	0.20	bcdgiopqrs	0.19	0.25	0.17	0.34	0.19	dfhinoruvwxBC
R&K	0.43	0.47	0.43	0.32	0.40	t	0.41	0.46	0.41	0.35	0.41	D
FED	0.29	0.34	-0.28	0.16	0.29	m	0.24	0.28	-0.21	0.19	0.24	pq
T&S	0.23	0.28	-0.21	0.15	0.23	s	0.21	0.25	-0.16	0.14	0.21	yB
HAN	0.21	0.25	-0.17	0.15	0.21	bcdopq	0.18	0.22	-0.11	0.14	0.18	bdu
JEF	0.23	0.27	-0.19	0.14	0.23	s	0.19	0.23	-0.09	0.11	0.19	bcdfuv
H&B-A	0.20	0.24	-0.13	0.14	0.20	bor	0.17	0.22	-0.02	0.12	0.17	cefhnvw
H&B-B	0.17	0.21	0.02	0.14	0.17	efghijr	0.20	0.26	0.14	0.11	0.20	bdgiouxyzBC
HONG	0.52	0.55	-0.52	0.14	0.52	u	0.48	0.51	-0.47	0.12	0.48	E
GRIG	0.24	0.29	-0.22	0.16	0.24	a	0.22	0.26	-0.18	0.16	0.22	szC
C&P	0.28	0.32	-0.26	0.16	0.28	n	0.25	0.29	-0.22	0.19	0.25	tA
PREV	0.21	0.25	-0.16	0.16	0.21	cp	0.17	0.21	-0.10	0.19	0.17	eghinowx
P&O	0.22	0.26	-0.16	0.15	0.22	dq	0.17	0.21	-0.04	0.15	0.17	hnw
Null	0.16	0.20	0	NA	0.16	efhjl	0.17	0.20	0	NA	0.17	mnouvwx

• NA = not available; Null = null model; models followed by the same letter did not differ in the Willcoxon Signed Rank test at the 0.05 level with the Holm correction for multiple testing.

REC curve

In the evaluation using the REC curve, the models were separated into different graphs for easier visualization (Figure 4A and B). The 12 best models are presented in Figure 4A and the remainder in Figure 4B. Table 4 also presents the values of AOC calculated from squared error (AOC_square) and absolute error (AOC_abs), and the results of the comparisons of the REC curves by the Wilcoxon test at the 0.05 level with the Holm correction for multiple testing.

Figure 4
REC (regression error characteristic) curve for the 25 pedotransfer functions (PTFs) tested for data set with depths of up to 30 cm: (A) 12 best functions: M&J-B, HUNT-B, ALEX, M&J-A, T&H, BEN-A, BEN-B, BEN-C, H&B-A, H&B-B, PREV and P&O (B) other PTFs. The red dashed line represents the accuracy of 50 % that can be used to visualize the error of the median.

Among the 12 best models, the Wilcoxon test showed that ALEX, BEN-B, H&B-B, M&J-A and T&H models presented a distribution similar to the null model. Otherwise, HUNT-B, H&B-A, BEN-A, PREV, and P&O models presented a different distribution from the null model, with an inferior performance for the majority of the error range (Figure 4A and Table 4). BEN-C and M&J-B also presented a different distribution from the null model, with a superior performance for a particular range of errors. The better performance of the M&J-B model is likely due to the variability of the data used in its generation (12,000 soil profiles) (Manrique and Jones, 1991). The result for BEN-C is probably related to similarity between data sets (Benites et al., 2007).

The 13 worst models presented a distribution dissimilar from the null model, with an inferior performance for the whole error range, showing that these models have low predictive ability (Figure 4B and Table 4). The HONG, R&K, and KAUR models had the worst results, with much poorer performance than the null model (Figure 4B and Table 4). De Vos et al. (2005) observed poor performance for both the KAUR and HUNT models using forest soil data from Belgium. These models were developed for very specific conditions, making them difficult to extrapolate (Table 2). Furthermore, the HUNT model used different methods to determine the ρb.

Nemes et al. (2010) emphasize that the model selected should be one developed from similar areas as regards soil genesis. However, even the PTFs developed for Brazilian soils, such as those generated by Benites et al. (2007), showed a performance slightly superior to the null model and for specific conditions.

The Wilcoxon test for REC curves according to depth showed similar results when considering all the data set (Table 6). For depths up to 10 cm (Figure 5A and B), ALEX, BEN-B, H&B-B, M&J-A, and T&H, together with M&J-B and LEO-A models, continued to present a distribution similar to the null model (Table 6). The BEN-C was the only model with a distribution different from the null model, but, again the gain in their application was very small and only for specific error ranges. The other five models, out of the 12 best ones HUNT-B, BEN-A, H&B-A, PREV, and P&O, presented a distribution different from the null model, with inferior performance for the majority of the error range (Figure 5A).

Figure 5
REC (regression error characteristic) curve for the 25 pedotransfer functions (PTFs) tested for soil data; A) 12 best functions for depths up to 10 cm: M&J-B, HUNT-B, ALEX, M&J-A, T&H, BEN-A, BEN-B BEN-C, H&B-A, H&B-B, PREV and P&O; B) Other PTFs for depths up to 10 cm; C) 12 best functions for depths of 10 to 30 cm: M&J-B, HUNT-B, ALEX, M&J-A, T&H, BEN-A, BEN-B BEN-C, H&B-A, H&B-B, PREV and P&O; D) Other PTFs for depths of 10 to 30 cm. The red dashed line represents the accuracy of 50 % that can be used to visualize the error of the median.

For depths of 10 to 30 cm (Figure 5C and D), the Wilcoxon test showed that 10 of the 12 best models did not differ from the null model (Table 6). A slight improvement occurred with the application of ALEX, M&J-B, BEN-B, and BEN-C models, but also for a particular error range and, therefore, cannot be interpreted as better performance of these models compared to the null model (Figure 5C). Previous studies have shown the influence of soil depth on the prediction of ρb. De Vos et al. (2005) found a 24 % improvement in the performance of PTFs in RMSE for ρb prediction at greater depths. Heuscher et al. (2005) noted that depth was responsible for approximately 1 % of the variation in ρb, and the greatest variation observed was 7 %. Benites et al. (2007) did not observe an improvement in the accuracy of the PTFs after separating the soils by depth (0 to 30 cm and 30 to 100 cm). Tranter et al. (2007) found better results when using the depth expressed on a logarithmic scale. Nemes et al. (2010) observed a decrease in the bias after separating the soil by depth.

In general, only BEN-C presented a slight improvement in ρb estimates for all situations (all data set, depths up to 10 cm and depths of 10 to 30 cm), when compared to the null model. The number of models with better results for topsoil was more restricted than for the subsurface. Having a worse performance for the topsoil as compared with the subsurface soil raises concerns as regards the estimation of soil C stock in which 0 to 30 cm are commonly considered as a reference layer (IPCC, 1997). Given the higher concentration of OC in the surface compared to the lower layer (Table 3), the error will be greater where it matters most.

The method used to determine the predictors of a PTF or the estimated variable (ρb) also seems to affect their performance. Both ρb and OC (or OM) can be measured by different methodologies (Sleutel et al., 2007; Blake and Hartge, 1986). The LOI and wet- oxidation methods are the two most commonly used methods for quantifying OM. However, the LOI method has no standard protocol and involves potentially confounding factors (Hoogsteen et al., 2015) even if the granulometric data could present variations since national and international classification systems often use quite different particle size ranges. In fact, Wösten et al. (2001) pointed out that there is no single source of variability, either PTF-related or soil-related, that can explain the uncertainty in every calculated functional aspect of soil behavior. The authors suggest using large and reliable data sets as well as PTFs developed from soils with similar attribute ranges to those used for the predictions. Here, we could see that similar ranges of soil attributes are not a guarantee of good predictions. The REC curve allows us to show that the 12 PTFs with the best indices presented, in general, a performance similar to the null model. BEN-C and M&J-B models presented the best results. The improvements observed were insignificant and were found only in specific error ranges.

Conclusions

The 25 models tested performed differently when observed and estimated bulk density values are compared, and the BEN-C and M&J-B models presented the best results.

The separation of data into two layers according to depth (0-10 cm and 10-30 cm) demonstrated a worse performance for the 0-10 cm layer, which raises concerns about the estimation of soil C in the upper layers that contain most of the organic matter.

The use of the REC curve as a form of analysis allowed for a simple and visual evaluation of the performance of the models.

The pedotransfer functions tested in this study showed a performance close to that of the null model (mean value) when estimating bulk density for soils from different regions of Brazil, indicating little or no additional benefit from the use of the null model.

Acknowledgements

We thank FAPESP (São Paulo Research Foundation) for financial support (Registry numbers: 2015/06804-0) and CNPq (Brazilian National Council for Scientific and Technological Development) for productivity grants in research to Eduardo Delgado Assad.

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Edited by

Edited by: Silvia del Carmen Imhoff

Publication Dates

• Publication in this collection
  Jan-Feb 2018

History

• Received
  05 Sept 2016
• Accepted
  21 Dec 2016