A Scale-Invariant Hyetograph Model for Stormwater Drainage Design Ke-Sheng Cheng and En-Ching Hsu Department of Bioenvironmental Systems Engineering National Taiwan University Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

The Role of A Hyetograph in Hydrologic Design Rainfall frequency analysis Design storm hyetograph Rainfall-runoff modeling Total rainfall depth Time distribution of total rainfall Runoff hydrograph Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Characteristics of Storm Hyetographs Although the shapes of storm hyetographs vary significantly, many studies have shown that dimensionless hyetographs are storm-type specific (Huff, 1967; Eagleson, 1970). In general, convective and frontal-type storms tend to have their peak rainfall rates near the beginning of the rainfall processes, while cyclonic events have the peak rainfall somewhere in the central third of the storm duration. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Representation of a Storm Hyetograph Rainfall depth process Dimensionless hyetograph Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Design Storm Hyetograph Models Duration-specific hyetograph models Keifer and Chu, 1957; Pilgrim and Cordery, 1975; Yen and Chow, 1980; SCS, 1986. Koutsoyiannis and Foufoula-Georgiou (1993) presented evidence that dimensionless hyetographs are scale invariant. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Objective of the Study The goal of this study is to propose a hyetograph modeling approach that have the following properties: Representative of the dominant storm type (storm-type-specific); Allowing translation between storms of different durations (scale-invariant); Characterizing the random nature of rainfall processes; Having the maximum likelihood of occurrence. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Selecting Storm Events for Hyetograph Design If rainfall data of both types were simultaneously utilized in order to develop design storm hyetographs, quite likely an average hyetograph results which characterizes the temporal rainfall variation of neither storm type. Selecting the real storm events that gave rise to the annual maximum rainfalls, the so-called annual maximum events, to develop design hyetographs. Annual maximum events tend to occur in certain periods of the year and tend to emerge from the same storm type. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Annual maximum rainfall data in Taiwan strongly indicate that a single annual maximum event often is responsible for the annual maximum rainfall depths of different design durations. In some situations, single annual maximum event even produced annual maximum rainfalls for many nearby raingauge stations. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Annual Maximum Events Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Selecting Storm Events for Hyetograph Design Using only the annual maximum events has two advantages: to focus on events of the same dominant storm type, to develop the design storm hyetographs using largely the same annual maximum events that are employed in constructing the IDF curves. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Simple Scaling Model for Storm Events – Instantaneous Rainfall Let represent the instantaneous rainfall intensity at time t of a storm with duration D. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Simple Scaling Model for Storm Events – Incremental & Cumulative Rainfall Incremental rainfall Cumulative rainfall Total rainfall Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Simple Scaling Model for Storm Events – Incremental & Cumulative Rainfall , h(t,D), and h(D,D) all have the simple scaling property with scaling exponent H+1, i.e., Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

IDF Curves and the Scaling Property The event-average rainfall intensity of a design storm with duration D and recurrence interval T can be represented by From the scaling property of total rainfall Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

IDF Curves and Random Variables is a random variable and represents the total depth of a storm with duration D. is the (1-p)th quantile (p =1/T) of the random variable, i.e.,   Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Random Variable Interpretation of IDF Curves Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

IDF Curves and the Scaling Property Horner’s Equation: D >> b , particularly for long-duration events. Neglecting b C = - H Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Theoretical Basis for Using Dimensionless Hyetographs in view of the simple scaling characteristics, the normalized rainfall rates of storms of different event durations are identically distributed. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Theoretical Basis for Using Dimensionless Hyetographs Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Gauss-Markov Model of Dimensionless Hyetographs Assume that the process {Y(i): i = 1, 2, …, n} is a Gauss-Markov process. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Gauss-Markov Model of Dimensionless Hyetographs By the Markov property, Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Modeling Objectives An ideal hyetograph should not only access the random nature of the rainfall process but also the extreme characteristics of the peak rainfall. Our objective is to find the hyetograph {yi , i = 1,2,…, n} that Maximize lnL, and y*: peak rainfall rate, t*: time-to-peak Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Lagrange Multiplier Technique The objectives can be achieved by introducing two Lagrange multipliers  and m , and minimizing the following expression: Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling SIGM Model System Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Model Applications Two raingauge stations in Northern Taiwan. Annual maximum events that produced annual maximum rainfall depths of 6-, 12-, 18-, 24-, 48-, and 72-hr design durations were collected. All event durations were divided into twenty-four equal periods i (i=1,2,…,24, D = event duration, =D/24). Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Parameters for Distributions of Normalized Rainfalls. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Evidence of Nonstationarity In general, Autocovariance function of a stationary process: For a non-stationary process, the autocovariance function is NOT independent of t. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Calculation of Autocorrelation Coefficients of a Nonstationary Process The lag-k correlation coefficients = correl.(Y(i), Y(i-k)) of the normalized rainfalls were estimated by Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Normality Check for Normalized Rainfalls by Kolmogorov-Smirnov Test Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Significance Test for Lag-1 and Lag-2 Autocorrelation Coefficients If = 0, then has a t-distribution with (N-2) degree of freedom. = autocorrel (Y(i), Y(i-k)) At significance level , the null hypothesis is rejected if . Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Significance Test for Lag-1 and Lag-2 Autocorrelation Coefficients Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Significance Test for Lag-1 and Lag-2 Autocorrelation Coefficients Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

SIGM Hyetograph - Hosoliau Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

SIGM Hyetograph-Wutuh Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Other Hyetograph Models Average Rank Model (Pilgrim and Cordery, 1975) Triangular Hyetographs (Yen and Chow, 1980) Alternating Block Approach (IDF-Based) Peak-Aligned Approach (Yeh and Han,1990) Clustering Approach (TPC) Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Hyetographs of TPC’s Clustering Approach Three major clusters account for 94% of the total events. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Hyetographs of TPC’s Clustering Approach Average hyetograph of the three major clusters (94% of total events). Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Model Evaluation Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling SIGM Model Validation Over thirty years of annual maximum events were utilized. Results were also compared against other hyetograph models. Validation parameters Peak rainfall rates Peak flow rates Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Event Validation: 54-08-18 Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Event Validation: 63-09-15 Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Event Validation: 69-08-27 Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Event Validation: 77-09-16 Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Comparison of Peak Rainfall Rates Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Comparison of Peak Flows Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Conclusions The SIGM hyetograph is the most suitable hyetograph model for the study area. Overall, it yields lowest RMSE of peak rainfall and peak flow estimates and its performance is more consistent across all gauges than other models. Although development of the alternating block and average rank models are computationally easier than the SIGM model; application of these two models may encounter difficulties. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University

Laboratory for Remote Sensing Hydrology and Spatial Modeling Conclusions The average rank model is duration-specific and requires rainfall data of real storms; however, gathering enough storms of the same duration may not always be possible. Although it dose not require rainfall data of real storms, the alternating block model is dependent on both duration and return period, and many hyetographs may need to be developed for various design storms. The SIGM hyetograph is storm-type-specific, scale-invariant and a unique hyetograph can be easily applied to design storms of various durations. Laboratory for Remote Sensing Hydrology and Spatial Modeling Department of Bioenvironmental Systems Engineering, National Taiwan University