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<br />II <br />il <br />I <br />II <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br />I <br /> <br />PROBABll.ITY OF FAlLURE <br /> <br />Risk is simply the probability of failure multiplied by the associated damages. Risk is then <br />integrated over the entire distribution of probabilities. The integration is simplified by the fact <br />that stream failures tend to occur once a threshold eVent is surpassed. So, below the threshold <br />event the probability of failure is near zero (giving zero risk) and above the threshold the <br />damages are fairly constant having mostly occurred once the threshold was exceeded. So the <br />probability of failure can be defined by the probability of the threshold event. To account for <br />existing, induced, and local hazards a factor (hazard weighting factor, HWF) is used to adjust the <br />probability of the threshold event, resulting in the following equation: <br /> <br />Risk - Probability * HWF * Losses <br />failure <br /> <br />EQ.ll <br /> <br />The hazard weighting factor (HWF) is the product of reach scale factors (RHWF) and local <br />scale factors (LHWF). The reach stability factor is a function of stream classification, and the <br />ratio of encroachment to the base active channel width, where: <br /> <br />RHWF _ Wb <br /> <br />where () .::;. 1 <br /> <br />EQ.12 <br /> <br />cr- <br /> <br />Base Width - Encroachment <br />Base Width <br /> <br />The weighting factor increases as the encroachment factor decreases (as encroachment <br />increases), This relationship shows that slight encroachments with little effect on the <br />geomorphically active area have little effect on reach stability, while a large encroachment <br />decreases reach stability. The exponent "b" depends on stream classification with the exponent <br />increasing as uncertainty in the stream form increases. Table 2 provides the recommended <br />exponent values: <br /> <br />Table 2. Reach Stability, HWF Exponent <br /> <br />Channel <br />Classification <br />ST-1 <br />ST-2 <br />ST-3a <br />ST-3b <br />ST-4 <br />ST-5 <br /> <br />Exponent, b <br /> <br />0.08 <br />0.34 <br />0.68 <br />0.77 <br />0.86 <br />1.10 <br /> <br />The weighting factors for local hazards are available for selected conditions. Cotton (1995) <br />compiles lateral and vertical weighting factors for local scour at bridges, and for river <br />countermeasures including spurs, guide bank, grade control, and bank revetments.t <br />14 <br />