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Frequency and duration curves

General

The frequency and duration curve option of hymosincludes the computation and plotting of:

• frequency curves,
• duration curves,
• average duration curves.

A convenient way to show the variation of hydrological quantities through the year, by means of frequency curves, where each frequency curve indicates the magnitude of the quantity for a specific probability of non-exceedance. The duration curves are a ranked representation of these frequency curves. The average duration curve gives the average number of occasions a given value was not exceeded in the years considered. The computation of the curves is elaborated in the next section. In the last section of this paragraph, the data requirements are dealt with.

Computational procedure

The frequency and duration curves are computed in the following way.

Frequency curves

A data matrix X_i,j for i = 1, m and j = 1, n is read from the data base covering m years (or months or days) of data with n elements per year (month, day). For each j=j₀ the data X_i,j0, i = 1, m is ranked and stored in R_i. Hence the length of vector R is m if there are no data missing, else the number m_eff. The probability that the i'th element of vector R_i is not exceeded is:

F_i = i/(m_eff +1)

A frequency curve connects all values of the quantity for j = 1, n with the common property of equal probability of non-exceedance. Generally a group of curves is considered which represents specific points of the cumulative frequency distribution for each j. Let the frequency curves be represented by matrix B~(k,j)_{, (k = 1, n}f~; j = 1, n) where n_f is the number of frequencies for which curves are created, then values for B_k,j0 are obtained by linear interpolation:

B_k,j0 = R_i-1 + (R_i -R_i-1 )(FD_k - F_i-1 ) / (F_i - F_i-1 )

where:
FD_k = frequency value of curve k

In case at j = j₀ insufficient data are available for interpolation, B_k,j0 is set equal to B_{k,j0-1. If j}0~ =1, B_k,j0 is set equal to zero.

Duration curves

When the data matrix B_k,j , k=1,n_f and j = 1, n is ranked for each k, the ranked matrix represents the duration curves for given probabilities of non-exceedance.

Average duration curves

By considering all data X_i,j, i = 1, m and j = 1, n the average duration curve indicates how often a given level of the quantity considered will not be exceeded in a year (or month or day) on the average; note that this definition differs from the meaning of the 50% duration curve! Exceedance levels FR~k~ are obtained by partitioning the range between X_max and X_min into 30 parts.
Let N_eff be the total number of data X_i,j > X_min , where N_eff ≤ m.n (depending on the number of missing data, and N_k the number of data not exceeding level FR_k, then the average duration AD_k of the event X_i,j ≤ FR_k reads:

AD≤k≤ = (N≤k≤ /N≤eff≤).n

Duration curves

When the data matrix B_k,j , k=1,n_f and j=1,n is ranked for each k, the ranked matrix represents the duration curves for given probabilities of non-exceedance.

Average duration curves

By considering all data X_i,j ,i = 1,m and j = 1,n the average duration curve indicates how often a given level of the quantity considered will not be exceeded in a year (or month or day) on the average; note that this definition differs from the meaning of the 50% duration curve! Exceedance levels FR_k are obtained by partitioning the range between X_max and X_min into 30 parts.
Let N_eff be the total number of data X_i,j > X_min, where N_eff £m.n (depending on the number of missing data, and N_k the number of data not exceeding level FR_k , then the average duration AD_k of the event X_i,j ≤ FR_k reads:

AD_k = (N_k /N_eff ).n

Data requirements

Series codes

Series can be selected by clicking the series in the 'series codes' list box. Only one series may be selected at a time.

Select Time interval of Analysis

The analysis timestep must always be greater than the selected series timestep. When an analysis timestep is selected HYMOS calculates the number of analysis elements in the specified period and the number of series elements within each analysis element.
The number of elements in the year, month or day: i.e. any period within the base period can be changed (e.g. if the base period is year and one wants to compute the frequency curves of daily water levels in April, then the start of the series should be 1 April of year xx and the number of elements in the year = 30),

Frequencies

The non-exceedance frequencies for which the frequency curves are requested should be entered. There is an option to enter 5 different frequencies.

Graph Output

When you want to present a graph of the frequency-durations, a graph option should be selected. Choose 'Frequency Curves', press Execute and press Graph from the functions Tab to show a Frequency Curves graph. Do the same for the other graph options.