Rainfall Frequency & IDF Analysis Worksheet
Continuous Multi-Duration & Disaggregation Modeling
Detailed Statistical Analysis
Viewing Duration:
| Sample Statistical Parameters ($n$ = 0) | |
|---|---|
| Mean ($\bar{X}$) | 0.00 mm |
| Std. Dev ($S$) | 0.00 mm |
| Skewness ($C_s$) | 0.00 |
| Variation ($C_v$) | 0.00 |
| Median | 0.00 |
| Min / Max | 0 / 0 |
| Std. Error | 0.00 |
| Range | 0.00 |
Goodness of Fit Evaluation
| Distribution | RMSE (mm) | Rank |
|---|
Detailed Predictions for 24-Hour (mm)
| Return Period ($T$) | Normal | Log-Normal | Gumbel (EV1) | Pearson III | Log-Pearson III |
|---|
* Note: Gumbel (EV1), Pearson III, and Log-Pearson III are the primary standard distributions used in hydrology for extreme events. Normal and Log-Normal are included primarily as theoretical baselines.
Probability Plot for 24-Hour Storm
Combined Depth Summary (mm)
Intensity-Duration-Frequency (IDF) Curves
IDF Empirical Equations
$I = \frac{A}{(T_c + B)^C}$
| Return Period ($T$) | Parameter $A$ | Parameter $B$ (min) | Parameter $C$ | R² Score |
|---|
* Parameters derived using iterative log-linear regression optimization. $I$ = Intensity (mm/hr), $T_c$ = Time of Concentration (minutes) acting as storm duration for peak design calculations (Kuichling, 1889; FHWA, 2002; Chow et al., 1988).
Theoretical Background
1. Empirical Plotting Position
Observed annual maximum data points are ranked from largest ($m=1$) to smallest ($m=n$). Their empirical return periods are assigned using the Weibull formula:
$$T = \frac{n+1}{m}$$
2. General Frequency Equation
All theoretical methods calculate the design magnitude ($X_T$) for a specific return period ($T$) using the general frequency equation (Chow, 1951):
$$X_T = \bar{X} + K_T \cdot S$$
Where $\bar{X}$ is the sample mean, $S$ is the sample standard deviation, and $K_T$ is the frequency factor which depends on the chosen probability distribution and the return period.
3. Probability Distributions
- Normal Distribution: Assumes a symmetrical bell curve ($C_s = 0$). The frequency factor $K_T = z$, where $z$ is the standard normal deviate for the probability $p = 1/T$. Rarely fits extreme precipitation well but serves as a theoretical baseline.
- Log-Normal Distribution: Applies the Normal distribution framework to the natural logarithms of the data ($y_i = \ln(x_i)$). This strictly prevents negative rainfall predictions and naturally accommodates the positive skewness typical of hydrologic extremes: $X_T = \exp(\bar{y} + z \cdot S_y)$.
- Gumbel (Extreme Value Type I): Specifically developed for modeling maximum extremes. It assumes a constant fixed skewness of $C_s \approx 1.139$.
The reduced variate is: $y_T = -\ln\left[-\ln\left(1 - \frac{1}{T}\right)\right]$
Scale parameter: $\alpha = \frac{\sqrt{6}}{\pi} S$
Location parameter: $u = \bar{X} - 0.5772 \alpha$
Magnitude: $X_T = u + \alpha y_T$ - Pearson Type III: A highly flexible three-parameter Gamma distribution that incorporates the actual sample skewness coefficient ($C_s$). The frequency factor $K_T$ is approximated using the Wilson-Hilferty transformation:
$$K_T = z + (z^2 - 1)\frac{C_s}{6} + \frac{1}{3}(z^3 - 6z)\left(\frac{C_s}{6}\right)^2 - (z^2 - 1)\left(\frac{C_s}{6}\right)^3 + z\left(\frac{C_s}{6}\right)^4 + \frac{1}{3}\left(\frac{C_s}{6}\right)^5$$
- Log-Pearson Type III (LP3): The federal standard (Bulletin 17C) for flood frequency analysis. It applies the Pearson Type III methodology to the base-10 logarithms of the data ($Y = \log_{10}X$). The predicted log-magnitude is $Y_T = \bar{Y} + K_T \cdot S_Y$, and the final rainfall depth is $X_T = 10^{Y_T}$.
4. Uncertainty and 95% Confidence Intervals
The uncertainty of the fitted distribution is estimated using the Standard Error ($S_e$). For the Best-Fit distribution, the standard error is calculated based on sample size ($n$), standard deviation ($S$), frequency factor ($K_T$), and skewness ($C_s$):
$$S_e = \frac{S}{\sqrt{n}} \sqrt{1 + K_T C_s + \frac{K_T^2}{2}(1 + 0.75 C_s^2)}$$
The 95% Confidence Interval band is then plotted as $X_T \pm 1.96 \cdot S_e$ (adapted for logarithmic bounds on Log-Normal and LP3). The bands widen significantly at high Return Periods due to extrapolation beyond the sample size record.
5. Goodness of Fit Evaluation (RMSE)
To determine the best-fitting distribution, the Root Mean Square Error (RMSE) is calculated between the observed historical data ($X_{obs}$) and the theoretically predicted values ($X_{pred}$) for the same empirical Weibull return periods:
$$RMSE = \sqrt{\frac{1}{n} \sum_{i=1}^{n} (X_{obs,i} - X_{pred,i})^2}$$
The distribution yielding the lowest RMSE provides the closest mathematical fit to the historical dataset and is automatically designated as the "Best Fit" method for generating the Combined IDF curves.
6. Continuous IDF Interpolation (Multiple Durations)
When multiple storm durations are provided, Intensity-Duration-Frequency (IDF) curves are generated. Depths are converted to intensities ($I = \text{Depth} / \text{Duration in Hours}$). For intermediate durations ($t$) between known durations ($t_1$ and $t_2$), Log-Linear Interpolation is used to produce continuous smooth lines:
$$\log(I) = \log(I_1) + (\log(I_2) - \log(I_1)) \cdot \frac{\log(t) - \log(t_1)}{\log(t_2) - \log(t_1)}$$
7. Temporal Disaggregation (Single Duration Mode)
In many regions, only daily (24-hour) rainfall data is reliably available. Short-duration design intensities can be mathematically estimated using empirical disaggregation models. The available methods in this tool are:
- Mononobe Equation: Applies a generalized power-law scale based on a user-defined exponent ($n$).
$P_t = P_{base} \cdot (t / t_{base})^n$ - Indian Meteorological Department (IMD): Uses the standard Mononobe power-law with a fixed empirical exponent of $n = 1/3$.
- NRCS/SCS Type II Distribution: Extracts the theoretical maximum rainfall fraction corresponding to each specific duration directly from the standard SCS Type II 24-hour cumulative mass curve.
- Custom Multipliers: Direct application of user-specified ratios (e.g., Bell's Ratios or localized regional studies).
8. IDF Empirical Equations
To conveniently represent the IDF relationship mathematically, an empirical three-parameter rational equation is commonly fitted to the derived intensities for each return period:
$$I = \frac{A}{(T_c + B)^C}$$
Where $I$ is the rainfall intensity, $T_c$ is the Time of Concentration in minutes (which is set equal to the storm duration for peak design calculations), and $A$, $B$, and $C$ are regional fitting parameters. To find the best fit, this tool iterates through values of $B$ and applies least-squares linear regression to the logarithmic transformation of the equation:
$$\ln(I) = \ln(A) - C \cdot \ln(T_c + B)$$
The combination of parameters that yields the lowest Sum of Squared Errors (SSE) and highest coefficient of determination ($R^2$) is selected.
References & Sources
- Chow, V. T., Maidment, D. R., & Mays, L. W. (1988). Applied Hydrology. McGraw-Hill. [Source]
- Federal Highway Administration (FHWA). (2002). Highway Hydrology (Hydraulic Design Series No. 2). Chapter 7 explicitly dictates setting Storm Duration = Time of Concentration for maximum design intensity. [Download PDF]
- Kuichling, E. (1889). The relation between the rainfall and the discharge of sewers in populous districts. Trans. ASCE. (The original paper establishing the Rational Method and the $T_c$ assumption).
- England, J. F., Jr., et al. (2019). Guidelines for Determining Flood Flow Frequency — Bulletin 17C. U.S. Geological Survey. [Download PDF]
- Gumbel, E. J. (1941). The Return Period of Flood Flows. Annals of Mathematical Statistics, 12(2), 163-190. [View Paper]
- Haan, C. T. (2002). Statistical Methods in Hydrology (2nd ed.). Iowa State Press. [View Book]
- Mononobe, N. (1936). Design of flood control structures. (Foundational basis for temporal rainfall disaggregation and intensity-duration scaling).
- NRCS/SCS (1986). Urban Hydrology for Small Watersheds (TR-55). (Standard Type II rainfall distributions for storm duration disaggregation).
- Indian Meteorological Department (IMD). (Standard empirical exponent of 1/3 for converting 24-hour rainfall to short-duration storm intensities).
- Weibull, W. (1939). A Statistical Theory of the Strength of Materials. Ingeniörsvetenskapsakademiens Handlingar, 151, 1-45. [View on Google Scholar]
No comments:
Post a Comment