Abstract
The design storm approach, where the subject criterion variable is evaluated by using a synthetic storm pattern composed of identical return frequencies of storm pattern input, is shown to be an effective approximation to a considerably more complex probabilistic model. The single area unit hydrograph technique is shown to be an accurate mathematical model of a highly discretized catchment with linear routing for channel flow approximation, and effective rainfalls in subareas which are linear with respect to effective rainfall output for a selected “loss” function. The use of a simple “loss” function which directly equates to the distribution of rainfall depth-duration statistics (such as a constant fraction of rainfall, or a ϕ-index model) is shown to allow the pooling of data and thereby provide a higher level of statistical significance (in estimating T-year outputs for a hydrologic criterion variable) than use of an arbitrary “loss” function. The above design storm unit hydrograph approach is shown to provide the T-year estimate of a criterion variable when using rainfall data to estimate runoff.
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Abbreviations
- A ω :
-
peak value (demand) of A for storm event ω
- A δω :
-
demand duringI δ based on functionF
- \(\alpha _{k_1 } \) :
-
timing offsets for channel link #1 used in the linear routing technique
- \(\alpha _{k_1 }^0 \) :
-
\(\alpha _{k_1 } \) corresponding to storm class\(\Omega _{\omega _0 } \)
- λ i jn :
-
effective rainfall proportion factors for subareaR j and stormi
- θ i jn :
-
effective rainfall timing offsets for subareaR j and stormi
- φ i j (s):
-
subarea unit hydrograph (UH) for subareaR j and stormi
- Ω:
-
probabilities space of all storms
- P :
-
probability measure on Ω
- \(\Omega _{\omega _0 } \) :
-
specific storm class
- ηi(·):
-
transfer function between measured effective rainfall and measured runoff, for stormi, using a Volterra integral model structure
- ηω(·):
-
transfer function for annual storm ω, usingF
- \(\eta _\omega ^{\omega _0 } ( \cdot )\) :
-
the restriction of ηω to the probability space\(\Omega _{\omega _0 } \)
- ηω(·):
-
ψω(·)W ω, ψω(·) a unit hydrograph
- a k 1 :
-
proportion factors for linear routing technique, used for channel link #1
- e ig (t) :
-
effective rainfall measured at the rain gage site, for stormi
- eg ω(·):
-
effective rainfall at the rain gage site for storm ω
- e ij (t) :
-
subareaR j effective rainfall for stormi
- e ω(·):
-
synthetic storm pattern input, usingF, for storm ω
- e δω (t) :
-
e ω (t) inI δ; 0, otherwise
- \(\bar e_\omega ^\delta (t)\) :
-
\(\bar I(e_\omega ^\delta ( \cdot ))\) fort inI δ; 0, otherwise
- \(\bar e_T^\delta \) :
-
T-year return frequency value of\(\bar I(e_\omega ^\delta ( \cdot ))\)
- Δe δω (t):
-
e δω (t)−\(\bar e_\omega ^\delta (t)\) inI δ, 0 outsideI δ
- \(E_{\omega _0 }^\delta ( \cdot )\) :
-
\(E(\Delta e_\omega ^\delta ( \cdot )|\bar I(e_{\omega _0 }^\delta ( \cdot )))\)
- ε δω :
-
Δe δω (·)−E δω (·)
- F :
-
effective rainfall function
- \(\bar I(e_\omega ^\delta ( \cdot ))\) :
-
mean value ofe δω (·) inI δ
- I δ :
-
peak storm pattern input time interval of duration δ; also, the operation of locating the peak duration δ of storm pattern input
- I(t) :
-
inflow hydrograph for linear routing
- M :
-
rainfall-runoff model
- 0(t) :
-
outflow hydrograph for linear routing
- \(\bar P_\delta \) :
-
mean precipitation for peak duration δ
- \(\bar P_T^\delta \) :
-
T-year return frequency value of\(\bar P_\delta \)
- P ω(·):
-
rainfall measured at the rain gage site, for storm ω
- P i (t) :
-
yeari annual storm precipitation
- Q ig (t) :
-
runoff hydrograph, for stormi, measured at the stream gage
- Q i (t) :
-
yeari annual storm runoff
- Q ω(·):
-
runoff hydrograph for storm ω
- Q δω (·):
-
runoff hydrograph resulting from peak time interval δ ofe ω(·)
- Q im (t) :
-
m-subarea link-node model output for stormi
- Q pT :
-
T-year return frequency peak flow rate
- q ij (t) :
-
runoff hydrograph from subareaR j, for stormi
- R :
-
total catchment
- R j :
-
subarea inR
- S T (t) :
-
T-year design pattern input
- S δ T (·) :
-
T-year design storm for peak time interval δ
- s,t :
-
temporal & integration variables
- x i :
-
annual storm event for yeari
- <k>:
-
vector notation for subscript sequence,k
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Hromadka, T.V., Whitley, R.J. The design storm concept in flood control design and planning. Stochastic Hydrol Hydraul 2, 213–239 (1988). https://doi.org/10.1007/BF01550843
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DOI: https://doi.org/10.1007/BF01550843