Most ptfs published in the literature are continuous pedotransfer functions (continuous ptfs), i.e. mathematical continuous functions between the water content at discrete values of potential or the parameters of a unique model of water-retention curve and the basic soil properties (mostly particle-size distribution, organic carbon content and bulk density) [12,17]. Besides these continuous ptfs that enable continuously the prediction of water content at particular water potentials [13] or estimation of the parameters of models of the water-retention curve [5,10,17], there are class pedotransfer functions (class ptfs) that received little attention, because their accuracy is considered as limited [15]. The existing class ptfs often provide average water contents at particular water potentials, or one average water-retention curve for every texture class [3,11]. Due to the range in particle-size distribution, clay mineralogy, organic matter content, and structural development within each texture class, water-retention properties for individual soils were considered as varying considerably [16]. Despite their possible inaccuracies, class ptfs enable the prediction based on successive stratification using soil characteristics. Moreover, class ptfs are easy to use because they require little soil information and are well adapted to the prediction of water-retention properties over large areas [9,15,16].

Class and continuous ptfs were developed using a set of 320 horizons, comprising 90 topsoils (from 0 to 30 cm depth) and 230 subsoil horizons (> 30 cm depth) collected in Cambisols, Luvisols, Planosols, Albeluvisols, Podzols, and Fluvisols [8] located mainly in the Paris basin and secondarily in the western coastal marshlands and Pyrenean piedmont plain. A set of 107 horizons comprising 39 topsoil and 68 subsoil horizons was constituted in order to test the ptfs established. These horizons were collected in Cambisols, Luvisols and Fluvisols [8] located in the South of the Paris basin. Basic characteristics and water-retention properties of the horizons were determined as described earlier by Bruand and Tessier [2] (Fig. 1, Table 1). Their bulk density (D_b) was measured by using cylinders 1000 cm³ in volume when the soil was near to field capacity.

In order to discuss the global validity of the ptfs, most studies used the root-mean-square error (RMSE), which is also called root-mean-square deviation or root-mean-square residual [17]. Because the RMSE varies according to both the prediction bias and precision, we computed the mean error of prediction (MEP) that enables discussion of the prediction bias alone, on the one hand, and the standard deviation of prediction (SDP) that enables discussion of the prediction precision alone, on the other hand. We computed MEP and SDP for the whole water potentials as follows:

In order to discuss the validity of the ptfs at the different water potentials, we computed also the mean error of prediction (MEP′) and the standard deviation (SDP′) of prediction at every water potential as follows:

The class ptfs developed in this note were established according to the texture (textural class ptfs) in the CEC triangle [4] and then according to both that texture and D_b (texturo-structural class ptfs). The resulting class ptfs corresponded to the average water content at seven water potentials, which was computed within every class of texture (textural class ptfs) (Table 2) and every class combining both texture and D_b (texturo-structural class ptfs) (Table 3). More complex class ptfs were established by fitting the van Genuchten's model [6] on the arithmetic mean value of θ at the different values of water potential using the RETC code [7] for every class of texture (VG texture class ptfs) according to the CEC triangle [4] and to the type of horizon (topsoil and subsoil) (Table 4).

Continuous ptfs were also developed. They correspond to multiple regression equations as follows:

The textural class ptfs underestimated very slightly the water retained (MEP = −0.003 cm³ cm⁻³) when they were applied to the test dataset without any other stratification than according to the texture. There was no decrease in the prediction bias with the texturo-structural class ptfs (MEP = −0.004 cm³ cm⁻³), but the bias was already very small with the textural class ptfs studied. However, the precision was slightly better with the texturo-structural class ptfs (SDP = 0.043 cm³ cm⁻³) than with the textural class ptfs (SDP = 0.045 cm³ cm⁻³) (Fig. 2a and b). Compared to the textural class ptfs, the VG textural class ptfs showed similar performance. The bias was very small (MEP = 0.002 cm³ cm⁻³) and the precision poor (SDP = 0.045 cm³ cm⁻³), as recorded for the textural class ptfs (Fig. 2c). The comparison of the class ptfs performance at every value of water potential showed small bias (−0.008 ≤ MEP′ ≤ 0.007 cm³ cm⁻³) except for θ_4.2 for the textural and texturo-structural class ptfs (MEP′ = –0.020 and –0.019 cm³ cm⁻³) and for θ_1.0 for the VG class ptfs (MEP′ = 0.014 cm³ cm⁻³), for which it was greater (Table 7). This comparison showed also poor precision for the three class ptfs studied, whatever the water potential (0.040 ≤ SDP′ ≤ 0.047 cm³ cm⁻³).

When applied to the test data set, the continuous ptfs lead to very small bias (MEP = –0.003 cm³ cm⁻³) and showed poor precision (SDP = 0.039 cm³ cm⁻³). Results showed a greater bias with the VG continuous ptfs (MEP = –0.008 cm³ cm⁻³) and similar poor precision (SDP = 0.039 cm³ cm⁻³) than with the continuous ptfs (Fig. 2d and e). The comparison of the continuous ptfs performance at every value of water potential showed small bias for the continuous ptfs (−0.006 ≤ MEP′ ≤ 0.005 cm³ cm⁻³), except for θ_4.2 (MEP′ = –0.022 cm³ cm⁻³). For the VG continuous ptfs, the bias was greater for six water potentials with absolute value of MEP′ ≤ 0.020 cm³ cm⁻³, except for θ_1.5 (MEP′ = 0.004 cm³ cm⁻³) (Table 7). The precision was poor for the simple and VG continuous ptfs (0.030 ≤ SDP′ ≤ 0.044 cm³ cm⁻³), but results showed that SDP decreased with the water potential.

Results showed very little difference between the ptfs studied. The bias recorded was small (–0.008 ≤ MEP ≤ 0.002 cm³ cm⁻³) and the greatest absolute value of bias was recorded with the VG continuous ptfs (MEP = –0.008 cm³ cm⁻³). On the other hand, the precision was poor (0.039 ≤ SDP ≤ 0.045 cm³ cm⁻³), the greatest precision being recorded with the two types of continuous ptfs studied. If the VG continuous ptfs led to the greatest precision (SDP = 0.039 cm³ cm⁻³), they led also the greatest bias value (MEP = –0.008 cm³ cm⁻³).