As described, in this research, six valid and widely used models for soil-moisture content curves were used. Fig. 2 shows the SWRC for all textures in the models evaluated during this study. By observing these figures, somewhat from a visual viewpoint,we can see the degree of conformance of each model with real data; however, statistical benchmarks were used for accurate examination. Based on the results obtained from the evaluation of the accuracy of the model stage, the highest amount of R2 among the field's data to the three models FX, BC, and VG in the silty clay loam texture with a value of 1 and the lowest value is 0.702 in the BC model was observed in the sandy loam texture. On the same basis, the highest amount of R2 was found among the lab's data of the FX model in clay with a value of 0.999 and the lowest value on the BC model in sandy clay loam texture with a value of 0.922.
In terms of AIC in the field's data, BL model in clay texture is the worst (with a value of -59.776), and LN model in silty clay loam texture is the best (with a value of -325.78) models, and among the lab's data BL model in silty clay loam texture is the worst (with a value of -38.381), and VG model in sand texture is the best (with a value of -2.51) models.
According to the results obtained from the MAD benchmark, the FX model in silty clay loam texture and BL model in the sand texture with a value of 2.718×10-11 and 0.299 known as the best and the worst model among field's data, respectively. Accordingly, the FX model in the clay texture and the BC model in the silt texture with the value of 0.0002 and 0.0218 was identified as the best and the worst model among the lab's data, respectively.
Investigating RMSE variations among the field's data showed that the maximum value of this benchmark belongs to the BL model in the sand texture (with a value of 0.036), which indicates the poor performance of this model in estimating the soil moisture curve in terms of this statistical benchmark. The lowest RMSE value on the FX model in the silty clay loam texture is 6.53×10-11, which indicates the superiority of this model compared to other models in terms of its accuracy. Among the lab's data, the highest value of this benchmark belongs to the BC model in the silt texture (with a value of 0.025), and the lowest RMSE on the FX model in clay texture is 0.0002. The result of this section on the low performance of the BC model in the RMSE benchmark corresponded to the results of Patil et al. [40].
A comparison of different values of MSE shows that among the field's data, the FX model in silty clay loam with a value of 4.426×10-21 is the most suitable and BL model in sand texture with 0.001 is the most unsuitable models for SWRC. Similarly, by comparing the values of this benchmark for the lab's data, the FX model in clay texture has the best (whit value of 8.299×10-8), and the BC model in the silt texture has the worst accuracy (whit value of 6.420×10-4).
The results of the GMER benchmark show that the DB model in the loam texture and BL model in the sand texture with a value of 1.00001 and 1.935 known as the best and the worst model among field's data, respectively. Accordingly, the LN model in the silty clay loam texture and the BC model in the silt loam texture with the value of 1.000008 and 1.211 was identified as the best and the worst model among the lab's data, respectively.
The best NS values in the field's data (similar to the R2) belong to the three models of FX, BC and VG in the silty clay loam texture with a value of 1, and the lowest value of this benchmark was 0.691 in the BL model in the silty clay loam texture was observed. Accordingly, similar to the results of the R2, the FX model in the clay data with the value of 0.999 best and the BC model in the sandy clay loam texture with a value of 0.920 showed the worst accuracy among the lab's data.
In the SMAPE benchmark among the field's data, the FX model in silty clay loam with a value of 8.237×10-11 is most suitable, and the BL model in the sand texture of 0.225 is the most unsuitable model for SWRC. Similarly, by comparing the values of this benchmark for the lab's data, the FX model in the clay texture (with a value of 7.477×10-4) has the best, and the BC model in the silt loam texture (with a value of 0.410) has the worst match with measurement data.
According to the results obtained from the OIF benchmark in the field's data, the DB model in the silty clay loam with a value of 0.010 is the best, and the LN model in the loamy sand texture with 0.197 is the worst match with the measurement data. In the lab's data, the LN model in clay texture (with the value of 0.197) has the best, and the LN model in the loamy sand texture (with the value of 0.277) has the worst conformance to the measurement data.
It is necessary to note that the process of computing the values of statistical benchmarks and fuzzy standardization for each texture has been implemented in all models, in which only the best and worst models are presented.
It seems that the best model in all benchmarks always is expected to have the best matching results, and vice versa about the worst model seems that have the worst matching results. However, in practice, in terms of some of them did not happen. For example, according to the results of calculating the BFM index, the DB model in the clay loam texture is presented as the best and most consistent model with the measurement data. While in terms of compliance with the statistical benchmarks (R2, AIC, MSE, RMSE, R, SMAPE) has an excellent quality, it has a good quality in three statistical benchmarks (MAD, NS, OIF) and has inferior quality in one of them (GMER). These results highlight the importance of simultaneously evaluating models in terms of different statistical benchmarks. Thus, simultaneous evaluation of different statistical benchmarks, such as those that occur in multi-criteria evaluation methods, allows the classification of models to vary according to the difference in response to each other relative to different statistical benchmarks. In the following, Table 4 shows the fitting accuracy of SWRC estimation models in each texture based on the numerical values of the BFM index. As can be seen, among the lab's data, the highest value of the BFM index belonging to the clay texture in the FX model, and the lowest of this index was observed on sandy loam in BC models. Furthermore, among the field's data, the highest BFM index belonged to clay loam texture in the BL model and the lowest in clay and BL models.
According to the classification given in Table 4, it is observed that among the lab's data, the BC model with 70% and the highest frequency of rank 6 (lowest rating) and DB model with 40% and the most abundance of rank 1 (top rank) are introduced as the worst and the best models, respectively. Moreover, among the field's data, the BC model with 50% and the highest frequency of rank 6 (lowest rank) and models of FX and DB each with 30% and the most abundant of rank 1 (highest rank), are introduced as the worst and the best models, respectively. The results of this section, in terms of the poor performance of the BC model, agree with the results of Giménez et al. [21], Nabizadeh and Beigi Harchegani [36], Ferreira et al. [16], and Bahmani and Ramazani [5]. It is worth mentioning that despite the introduction of the BC model among all lab and field data with the highest frequency of rank six as the most improper model, it cannot be concluded that this model does not have the proper performance in all texture because, as in Table 4 are found, in some textures have ranked 1, 2 or 3. Thus, considering the low rating of this model in most of the textures, it should be doubted about the application and accuracy of its results compared to other models. Since the BC model showed a slight accuracy in fitting, it can be stated that any size of soil pores is more uniform, by increasing the amount of λ, which indicates the pore size distribution, the slope of the SWRC increases in the non-saturated soil [5]. This model offers acceptable results for coarse-textured soils with even pores, but its results are unsuitable near the saturation point, especially in heavy textured soils (clay, clay loam, silty clay loam) [21]. According to the DB model, which among other models, with seven parameters in its formula, considers the maximum value of the parameters, it can be argued that using more parameters and considering more soil properties (but not always) can dramatically increase the conformance of the model results with the measured results. However, the ease of using the model is reduced [8].