2.3 Marco referencial
2.3.2 Etapas de desarrollo del Lenguaje
Annual variability of river flow was calculated by identifying the annual peak and minimum flows and plotting them against time. This was to show the annual variability for the same period (Knighton, 1984; Bengtson, 2009). Mean annual
discharges were plotted against time to show how mean is varying with time and with low flows and high flows (Henderson, 2007).
3.5.1 Stream Flow Duration Analysis
The flow duration curve is a plot that shows the percentage of time that flow in a stream is likely to equal or exceed some specified value of interest. It can be used to show the percentage of time river flow was expected to exceed a design flow of some specified value, or to show the discharge of the stream that occurs or is exceeded some percent of the time. The basic time unit used in preparing a flow- duration curve greatly affect its appearance. For most studies, mean daily discharges are used. These gave a steep curve. When the mean flow over a long period was used (such as mean monthly flow), the resulting curve becomes flatter due to averaging of short-term peaks with intervening smaller flows during a month (Overeem, 2009). Extreme values are averaged out more and more, as the time period gets larger (e.g., for a flow duration curve based on annual flows at a long-record station).
Step 1: Sort (rank) average annual discharges for period of record from the largest value to the smallest value, involving a total of n values.
Step 2: Assign each discharge value a rank (M), starting with 1 for the largest daily discharge value.
P = 100 * [M / (n + 1)] ………(6)
P = the probability that a given flow will be equaled or exceeded (% of time)
M = the ranked position on the listing (dimensionless)
n = the number of events for period of record (dimensionless)
3.5.2 Flood Frequency Analysis
The Log-Pearson Type III Distribution and Linear Regression Analysis of Discharge were applied in this research.
3.5.2.1 Log-Pearson Type III Distribution
Log-Pearson Type III distribution has found wide application in hydrologic analysis. The log-Pearson Type III distribution is a three-parameter gamma distribution with a logarithmic transform of the variable. It is widely used for flood analyses because the data quite frequently fit the assumed population. According to Cheng et al. (2006), using the mean, standard deviation, and skew coefficient for any set of log-transformed annual peak flow data, the flood with any exceedence frequency can be computed using equations:
𝑌̂ = log X = Y + KS y………(7)
Where 𝑌̂ is the predicted value of log X, 𝑌̅ and Sy and K is a function of the exceedence probability and the coefficient of skew.
The specific steps for making a basic log-Pearson Type III analyses without any of the optional adjustments are as follows:
1. Make a logarithmic transform of all flows in the series (Yi = log Xi).
Compute the mean (𝑌̅), standard deviation (Sy), and standardized skew (G) of the logarithms using Equations 8, 9, and 10 respectively. Round the skew to the nearest tenth.
𝑄̅ =
∑𝑛𝑖=1𝑄𝑖𝑛 ………(8)
where,
𝑄̅ = average or mean peak.
𝑆 = [
∑𝑛𝑖=1(𝑄𝑖−𝑄̅)2𝑛−1
]
0.5
……….……(9)
S, is defined as the square root of the mean square of the deviations from the average value.
𝐺 =
𝑛 ∑𝑛𝑖=1(𝑄𝑖−𝑄̅)3(𝑛−1)(𝑛−2)𝑆3……….(10)
G = the coefficient of skew
2.Since the log-Pearson Type III curve with a nonzero skew does not plot as a straight line, it was necessary to use more than two points to draw the curve.
The curvature of the line will increase as the absolute value of the skew increases, so more points will be needed for larger skew magnitudes.
3.Compute the logarithmic value 𝑌̂ for each exceedence frequency using Equation 10.
4. Transform the computed values of step 3 to discharges :
𝑋̂ = 10
𝑌̂………(11)in which 𝑋̂is the computed discharge for the assumed log-Pearson Type III population.
5.Plot the points of step 4 on logarithmic probability paper and draw a smooth curve through the points.
The sample data can be plotted on the paper using a plotting position formula to obtain the exceedence probability. The computed curve can then be verified, and, if acceptable, it can be used to make estimates of either a flood probability or flood magnitude.
3.5.2.2 Linear Regression Analysis of Discharge
Regression analysis was developed to detect the presence of a mathematical relation between two or more variables subject to random variation, and to test if such a relation, whether assumed or calculated, is statistically significant (Oosterbaan, 1994). If one of these variables stands in causal relation to another,
that variable is called the independent variable. The variable that is affected is called the dependent variable. Linear two-variable regressions are made according to one of two methods. These are the ratio method and the least squares method. This study adopted the ratio method because unlike least squares" method, it is used when the random variation increases or decreases with the values of the variables. If this is not the case, the least-squares method is used. The ratio method, as we use it here, consists of two steps, namely:
1. Calculate the ratio p = y/x of the two variables y and x;
Where x was taken to be stream flow when y is rainfall, x is water supply when y is annul flow and x is water supply when y is rainfall.
2. Calculate the average ratio pav, its standard deviation sPav, and its upper and lower confidence limits Pu and v, to obtain the expected range of in repeated samples.
In the ratio method, if the variation in the data (x, y) tends to increase linearly, the ratio method can be applied. This reads as equation 12 and 13:
y = p.x + ε or ŷ = p.x
………(12)Or
Y/x = p + ε' or (ŷ/x) = p
………(13)p = a constant (the ratio)
ŷ = the expected value of y according to the ratio method
ε and ε' = a random deviation
(Ŷ/x) = the expected value of the ratio y/x
The regression line requires that smoothening of data be done to remove errors and fill in missing discharge data. This was done using moving averages method. The degree of correlation is calculated by applying a coefficient of correlation to data concerning the two phenomena. The most common correlation coefficient is expressed as equation 14: ∑(𝑋 𝜎𝑥∙ 𝑌 𝜎𝑦) 𝑁 ………(14)
in which x is the deviation of one variable from its mean, y is the deviation of the other variable from its mean, and N is the total number of cases in the series. A perfect positive correlation between the two variables results in a coefficient of +1, a perfect negative correlation in a coefficient of -1, and a total absence of correlation in a coefficient of 0. Intermediate values between +1 and 0 or -1 are interpreted by degree of correlation.