4. Limitaciones generales
4.4. Concurrencia con otros derechos
Errors associated with fitting Waxman-Smits empirical relationship to actual laboratory observed data are presented in the form of root-mean-square error and percentage root mean square error. Data correlation is also presented as a means of quantifying laboratory data spread away from the model. Table 5.12 displays the error and correlation results of all the Waxman-Smits fitting curves to experimentally derived laboratory data. Figures 5.33 to 5.35 below presents the percentage RMS errors for the Waxman-Smit models associated with both the 5 cm and 1 cm square array resistivity measurements. Percentage RMS errors associated with orientation A vary between square array size utilised and between soil sample lithology. Troughs 1, 2, 4 and 5 (Fig 5.33) have %RMS errors all less than 60% for both square array sizes of measurements of orientation A. The 1cm square array records %RMS errors almost double those associated with the 5cm array and for trough 1 these are 57.1% and 31.4% respectively. Trough 3 follows suit, its 1cm, A oriented models record %RMS errors higher than their 5 cm counterpart. Errors associated with Trough 3 – both 5cm and
1 cm square arrays – are considerably higher than models for all other troughed soils (Trough 6; 5cm orientation A being the only exception). Percentage RMS errors orientation A of trough 6 trend differently to the other 5 troughs, as it displays %RMS errors higher for the 5cm array than 1cm array.
5 cm Square Array 1 cm Square Array
Trough 1 Average Orient. A Orient. B Average Orient. A Orient. B
RMS 29.68 8.47 55.65 17.41 16.30 19.45
%RMS 42.60 31.35 50.56 26.80 57.10 23.75
Correlation 0.89 0.77 0.88 0.91 0.83 0.94
Trough 2 Average Orient. A Orient. B Average Orient. A Orient. B
RMS 34.17 38.52 31.63 55.01 34.06 84.35
%RMS 21.51 34.72 18.48 30.88 56.68 35.68
Correlation 0.85 0.76 0.90 0.92 0.90 0.91
Trough 3 Average Orient. A Orient. B Average Orient. A Orient. B
RMS 12184.1 1755.01 22756.80 99151.3 64858.60 133553.0
%RMS 92.20 68.19 108.24 113.60 104.61 132.78
Correlation 0.90 0.87 0.91 0.87 0.87 0.87
Trough 4 Average Orient. A Orient. B Average Orient. A Orient. B
RMS 46.18 56.74 37.02 65.94 73.34 60.36
%RMS 26.97 33.66 22.79 83.47 58.71 205.39
Correlation 0.86 0.80 0.91 0.82 0.84 0.78
Trough 5 Average Orient. A Orient. B Average Orient. A Orient. B
RMS 103.03 60.62 145.81 210.81 60.67 347.97
%RMS 26.98 35.01 25.41 32.19 58.92 30.00
Correlation 0.85 0.84 0.85 0.87 0.73 0.89
Trough 6 Average Orient. A Orient. B Average Orient. A Orient. B
RMS 1653.11 203.87 2994.26 266.02 144.34 412.00
%RMS 35.81 76.00 41.79 52.38 48.98 84.12
Correlation 0.93 0.88 0.94 0.97 0.89 0.98
Trough 6 w/o
cracking Average Orient. A Orient. B Average Orient. A Orient. B
RMS 2008.86 203.87 3725.70 266.02 144.34 412.00
%RMS 31.49 76.00 33.90 52.38 48.98 84.12
Correlation 0.93 0.88 0.94 0.97 0.89 0.98
Table 5.12. Root-mean-square error, percentage root-mean-square error and correlation between laboratory measured resistivity and empirically modelled resistivity by Waxman-Smits equation. Error and correlation results for measurement orientations A and B for both 5cm and 1cm square arrays. Error and correlation results refer to modelling performed on the average of two orientations (A & B) for each square array.
Figure 5.33. Percentage RMS error of modelled results from measured resistivity results using measurement orientation A. Presented are model errors associated with both 5cm and 1cm square arrays.
Percentage RMS errors for models associated with experimental resistivity measurements in orientation B are shown in Figure 5.34. Waxman-Smit models of troughs 1, 2 and 5 show %RMS errors less than 50%, with troughs 2 and 5 recording errors half that. All troughs except trough 1 provide lower %RMS values for the 5cm square array than the 1cm. Trough 1 is the exception, its 5cm value being 50.6% and its 1cm being just under half that value at 23.8% RMS error. Troughs 3 and 4 display seemingly high %RMS errors when compared to those provided by the other troughs. Trough 3 has relatively high %RMS errors at 108.2% and 132.8% for the 5cm and 1cm square arrays respectively. Trough 4 display great variation in the %RMS error values of its two empirical models. The model produced using the 5cm square array has an error of 22.8%, however, the error pertaining to the 1cm array is 205.4%, almost a factor of 10 higher. This could be attributed to processes taking place in the soil that the Waxman- Smit model does not account for such as electronic conduction within ironstone clasts. For example, coarse sand to fine gravel size ironstone clasts were present in the sample of trough 4.
The Waxman-Smit models produced by passing the arithmetic mean of resistivities derived from orientation A and B measurement orientations to the model produce %RMS errors as presented in Figure 5.35. For all but trough 1 it is the model utilising 5cm array resistivities which reports lower %RMS errors than the equivalent 1cm array model.
Figure 5.34. Percentage RMS error of modelled results from measured resistivity results using measurement orientation B. Presented are model errors associated with both 5 cm and 1 cm square arrays.
The clean sand of Trough 3 again produces the largest errors with 5cm and 1cm giving errors of 92.2% and 113.6% respectively. Clays of the Whitby Mudstone Formation contained within troughs 1 and 2 offer some of the lowest %RMS errors, all being less than 43%. The greatest variability between models with different array sizes is trough 4 which shows a difference of over 56%. By removing the cracking affected resistivities from the siltstone of trough 6 reduced the 5cm model error by 4.3%.
Figure 5.35. Percentage RMS error of modelled results from mean averaged laboratory measured trough resistivity data. Averaged resistivity of the two orientations A and B for each square array size. Presented are model errors associated with both 5 cm and 1 cm square arrays.
Trough 3 %RMS errors appear high (between 68.2% and 132.8%) because Waxman-Smit equation did not produce a successful fit to the laboratory data. Observation of the apparent misfit between laboratory data and the model makes it clear that the model is not appropriately applied in light of trough 3 lithology. This could be due to the Waxman-Smit equation not being
the most appropriate electro-petrophysical equation for this sample. Trough 3 is a fine, clean sand and so Archie’s Law may be a more appropriate model to apply to this sample.