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89 CHAPTER 11 FUTURE HYDROGRAPHS AND SCOUR RISK ANALYSIS 11.1 BACKGROUND a given sequence of daily stream-flow values throughout the life, Lt, of the structure. The randomness of the hydrologic Since the SRICOS-EFA Method predicts the scour depth as forcing suggests combining the scour model with some hydro- a function of time, one of the inputs is the velocity versus time logical and statistical analyses. If the stream-flow sequence (or curve, or hydrograph, at the foundation location. This hydro- hydrograph) is modeled as a stochastic process, it is possible graph should cover the period over which the scour depth must to set up a Monte Carlo procedure that samples different real- be predicted. A typical bridge is designed for 75 years. There- izations of the hydrograph (of length Lt) from that process and fore, the design for a new bridge requires the knowledge of the estimates (using the SRICOS-EFA Method) the scour depth, hydrograph from the year of construction until 75 years later. d, at the end of the bridge life for each of them. Thus, d is The question is: how can one obtain the future hydrograph regarded as a random variable and its statistics can be studied covering that long period of time? This requires predicting the in detail to determine the risk of failure associated with differ- future over a 75-year period. ent choices of the design value of the scour depth. One solution is to use a hydrograph recorded at a nearby The modeling of daily stream-flow, Q, can be tackled using gage station over the last 75 years and assume that the future different approaches (e.g., Bras and Rodriguez-Iturbe, 1986; hydrograph will be equal to the past hydrograph. If the gage Montanari et al., 1997; 2000) corresponding to different levels is not at the future bridge location, the discharge can be mul- of complexity. A first simple analysis suggested here consid- tiplied by the ratio of the drainage area at the bridge site over ers Q as a random, uncorrelated variable. A suitable distribu- the drainage area at the gage site. If the record at the gage sta- tion is fit to the data and the hydrographs are then generated as tion is not 75 years long, one can simply repeat the recorded a series of values sampled from such a distribution. Ongoing hydrograph until it covers the 75-year period. If the recorded research also is applying other stochastic models to account for hydrograph does not include the design flood (100-year flood both the autocorrelation and the memory of the process and is or 500-year flood), one can spike the hydrograph with one or assessing whether the temporal structure (i.e., both autocorre- more of those floods before running the SRICOS program lation and memory) of the stream-flow sequences is able to (Figure 11.1). affect the statistical properties of the scour-depth probability Another solution is to use the new technique that is pre- distribution. sented here. This technique consists of using a past hydro- The theoretical distribution used to model daily stream-flow graph, preparing the frequency distribution plot for the floods observations needs to be defined only for positive values of Q, within that hydrograph, sampling the distribution randomly to have a positive skew, and to be able to provide an accurate and preparing a future hydrograph for the required period representation of the extreme values (i.e., good fit at the upper that has the same mean and standard deviation as the mea- tail of the distribution). As expected, the extreme values are sured hydrograph. This process is repeated 10,000 times and, found to greatly affect the scour depth estimates and an impre- for each hydrograph, a final scour depth (the depth reached cise modeling of stream-flow maxima could easily lead to after 75 years of flow) is generated. These 10,000 final depths unrealistic estimations of the scour depth statistics. Logarith- of scour are organized in a frequency distribution plot with a mic transformations are frequently used to study stream-flow mean and standard deviation. That plot can be used to quote extremes (e.g., Chow et al., 1988; Benjamin and Cornell, a scour depth with a corresponding probability of occurrence, 1970); therefore, a log-normal distribution can be a good can- or better, to choose a risk level and quote the corresponding didate for modeling the daily stream-flows. The method of final depth of scour. moments is used to determine the parameters of the distribu- tion. As such, Q is expressed as the exponential of a normally distributed random variable, y, with mean 11.2 PREPARATION OF THE FUTURE HYDROGRAPHS 1 Q2 y = Log (11.1) 2 Q 2 The SRICOS-EFA Method determines the scour depth at 1 + the end of the bridge life as a progressive process driven by Q