La inclusión como una alternativa a la diversidad

In order to check the extent of the predictive accuracy of this probabilistic method, the sediment yields that were computed using equation 5.2 were checked using the discrepancy ratio test at 50% exceedance probability. The technique compares all the predicted sediment yields against all the observed sediment yields using the discrepancy ratio, xi, whereby each predicted value is divided by the corresponding

actual observed value. The discrepancy ratio xi, should be a good indicator of the

predictive accuracy of the probabilistic approach in predicting the sediment yield.

In mathematical terms the discrepancy ratio would be given by the following relationship:

uvwxy

uvz{w

 |

Where

6kn} = Simulated sediment yield

6r~k = Observed sediment yield

The simulated sediment yields refer to the sediment yields calculated using equation 5.2. The simulated sediment yields using equation 5.2 were divided by their corresponding observed sediment yields in Appendix B for all the regions. The following relationship was obtained relating to the calculated value of xi.

56 0.33 ≤ n ≤ 3; 81% of the data was in this range 0.5≤ n ≤ 2; 68% of the data was in this range 0.67 ≤ n ≤ 1.5; 43% of the data was in this range

The ranges of the discrepancy ratios obtained in the current statistical approach were compared with those obtained in the previous statistical approach (Rooseboom et al., 1992) and the results were:

0.33 ≤ n ≤ 3; 70% of the data was in this range (Rooseboom et al., 1992) 0.5≤ n ≤ 2; 47% of the data was in this range (Rooseboom et al., 1992) 0.67 ≤ n ≤ 1.5; 32% of the data was in this range (Rooseboom et al., 1992)

The results for individual regions are shown in Table 5.6.3 for the probabilistic approach in this thesis.

Table 5.6.3 Discrepancy ratio results for the probabilistic method

These ranges are within the limits of acceptable predictive accuracy considering the complex nature of the spatial variability in sediment yield. However, these values have been computed at 50% probability of exceedance implying that a factor of one (1) has been adopted for all calculated sediment yield values. For higher or lower confidence bands, the multiplication factors from Appendix E are applied. Caution must be taken when applying these factors to avoid over prediction. The probabilistic methodology/approach appears to over predict very small observed sediment yields

Region Obs. n 0.67<xi<1.5 0.5<xi<2.0 0.33<xi<3.0 1 18 41 64 77 2 25 36 68 84 3 7 71 71 86 4 30 44 66 72 5 12 33 61 83 6 8 56 67 89 7 19 46 65 77 8 14 26 60 80 9 9 44 78 89

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and under predict high sediment yields at 50% probability of exceedance. This is evidenced by the relationship between the observed data and simulated data on the graphs in Appendix F. For example, the graph for region 1 in Appendix F shows higher calculated sediment yields in the vertical axis for corresponding low observed sediment yields in the horizontal axis. This is because the method is based on the general concept of regional sample median assumed at 50% probability of exceedance.

In essence, the estimation of the sediment yield is developed from the average of the observed data series which is taken as the 50th percentile. Since theoretically a percentile is the value of a variable below which a certain percent of observations fall (Wikipedia, 2009), the estimated median sediment yield value calculated by equation 5.2 gives a sediment yield value below which 50% percent of observed sediment yields fall. The probabilistic method developed around the 50th percentile value should typically over predict probably at least half of the sediment yields. The estimated median sediment yield calculated using equation 5.2 acts as a reference point whose main application is to provide a sediment yield value that has a 50% exceedance probability. This results in problems in regions where there is high variability in the sediment yield values from the lowest to the highest.

The standardised average yield itself may be already over predicting some small sediment yield in the region. This is the reason why data points on the graphs in Appendix F are characterised by poor scatter along the line of perfect fit. The probabilistic approach does not derive direct relationships between the observed and calculated sediment yields. The method calculates a value that statistically masks all values below it depending on the specified probability of exceedance. For example at 50% exceedance probability, the method calculates a value whereby almost 50% of the data in the original sample size would have been below it.

The results given in Appendix F show that for some low observed sediment yields, say less than 100t/km2.a, the method gives relatively higher simulated yields. Similarly, the higher observed sediment yields above the standardised average yield appear to be under predicted. In practice to avoid over predicting or under predicting,

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two possible measures could be undertaken. The first measure would be to check the predicted sediment value at 50% confidence band against the nearest observed yield value within the region and compare the results. Secondly, the graphs for the statistical distribution (probability of exceedance) of the observed sediment yields for each region, shown in Appendix D, can be used to compare the predicted value against the expected sediment yield value from the graph at any specific probability of exceedance. In other words, the probabilistic distribution of the observed sediment yields for each region gives an estimate of the general variation of the expected sediment yields within a given region.

Depending on the comparative results, an appropriate confidence band can be adopted. If the estimated median sediment yield is found to be lower than the comparative sediment yield, then the factors provided in the confidence bands’ graphs in Appendix E can be used depending on the preferred confidence band and applicable catchment area. The discrepancy ratio test outlined in Table 5.6.3 is considered a significant measure of the predictive accuracy of sediment yield prediction approaches in sedimentation engineering particularly where multiplication factors are applied to achieve higher confidence levels.