From hydrological processes to models of the Rainfall-Runoff transformation
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 1 Rainfall-Runoff transformation
Lecture content Skript: Ch. VI.1, VI.1.1, VI.3 – 3.2.2.5
– rationale for modelling the rainfall-runoff (R-R) transformation – introduction to rainfall-runoff models – runoff concentration concept – lumped rainfall-runoff models – unit hydrograph – synthetic unit hydrographs – R-R model parameter estimation
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 2 Rationale for modelling the rainfall-runoff transformation • The purpose of modelling the transformation of rainfall into runoff is to simulate the response of river basin to meteorological forcing ⤷ to solve design problems ⤷ to investigate the variability of hydrological processes and their impact on river flows ⤷ to replace missing data, to extend historical data, to overcome the shortcoming of limited measurements ⤷ to predict river flows in ungauged basins
⤷ support to engineering, design and decision making ⤶
ê models are characterised by different spatial and temporal representation of the R-R transformation depending on the purpose of modelling
SPATIAL SCALE TEMPORAL SCALE PROCESS REPRESENTATION ê ê ê • distributed • continuous • physically based • lumped • event-based • conceptual
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 3 Modelling the rainfall-runoff transformation – example 1 Problem: • given a river with limited amount of historical flow observations but long precipitation daily records
• what is the amount of water that can be derived from a river to satisfy water demand (QD) for irrigation and water supply? ⤷ generation of daily flows using a continuous ⤷ estimation of the flow duration curve and rainfall-runoff model analysis of its variability Q(t) ê Q(t) ê
QD t [days] 1 365 days the continuous R-R model accounts for all the hydrological processes contributing to the basin response: interception, evapotranspiration, infiltration, sub-surface flow, baseflow, surface flow ⤷ description of storm and interstorm processes ⤶ Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 4 Modelling the rainfall-runoff transformation – example 2 Problem:
• given a river with insufficient data to compute the flood peak for a given return period (QR) by statistical analysis
• what is the R-year return period flood discharge (QR)?
⤷ indirect estimation using an event-based rainfall-runoff model ⤶ ê synthetic flood DDF curve hyetograph Q(t) hydrograph H i(t) QR R-R model T t t the event-based R-R model accounts for the hydrological processes contributing to the flood response: infiltration, sub-surface flow, surface flow ⤷ description of storm processes ⤶ Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 5 The event-based rainfall-runoff transformation
observed data RAINFALL or
DDFs + synthetic hyetograph
e.g. SCS-CN INFILTRATION model
RUNOFF CONCENTRATION UNIT ê HYDROGRAPH basin response function
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 6 Rainfall-runoff transformation model assumptions nature of the physical processes of the model approximations rainfall-runoff transformation (assumptions) ê ê • non linear • linear
⤷ PR=50 years à QR=50 years ⤷ PR=50 years à QR=50 years
• time varying • time invariant ⤷ the basin response varies from storm to storm ⤷ the basin response is invariant for any storm
• distributed in space • lumped in space ⤷ rainfall and soil properties are variable in space ⤷ rainfall and soil properties are constant in space ê LINEAR, CONCEPTUAL, LUMPED MODELS OF THE RAINFALL RUNOFF TRANSFORMATION ê • the rainfall input is constant over the watershed (“average” in space) • the infiltration model is characterised by one parameter set, which describe the “average” infiltration response of the watershed • the runoff concentration model parameters do not change with changing rainfall input or watershed soil properties Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 7 Linear model of rainfall-runoff transformation (1)
transfer INPUT OUTPUT function
linear time invariant
• time-invariance à stationarity
⤷ if an input I1(t) produces an output O1(t)
⤷ if an input I2(t+τ) produces an output O2(t+τ) ⤷ two inputs shifted by τ produce two outputs which are also shifted by τ ⤶
• linearity à ① proportionality and ② addivity (superposition of the effects) ⤷ ① if an input I(t) produces an output O(t) ⤷ an input c⋅I(t) produces an output c⋅O(t), c=const. ⤶
⤷ ② if an input I1(t) produces an output O1(t) and an input I2(t) produces an output O2(t)
⤷ an input I1(t) + I2(t) produces an output O1(t) + O2(t) ⤶
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 8 Linear model of rainfall-runoff transformation (2) INPUT transfer OUTPUT p(t) function q(t)
linear time invariant
• under the condition of stationarity and linearity • if p(t) and q(t) are respectively the input (net rainfall) and the output (runoff) functions ⤷ it can be demonstrated that the response of the system to a continuous input p(t) can be treated as a sum of infinitesimal inputs ê the response q(t) can be written as solution of a linear system with constant coefficients ê d nq d n−1q dq p(t) = a0 + a1 + ...+ an−1 + anq dt n dt n−1 dt t CONVOLUTION which can be solved with q(0) = q0 = 0, q′(0) = q0′ = 0, ... as q t = h t − τ p τ dτ ( ) ∫ ( ) ( ) INTEGRAL 0 h(t-τ) is the basin response function, which describes the runoff concentration Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 9 Instantaneous Unit Hydrograph (IUH) - concept
δ(t) linear h(t) = response to δ(t), Dirac h(t) = 0 ∀t < 0 function transfer function
δ t − t = 0 ∀t ≠ t ( 0 ) 0 h(t) = 1 because of continuity ∞ ∫ δ t − t dt = 1 ∫ ( 0 ) −∞ δ(t) applied after τ à response shifted by τ à δ(t-τ) à h(t-τ)
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 10 Instantaneous Unit Hydrograph (IUH) - application
p(t)
pulse of length dτ and • intensity p(τ) the infinitesimal response of the system, dq(t), is given by the product of the area of the impulse, p(τ)⋅dτ, and the value of the t unitary response function at t-τ, h(t-τ) dτ ⤷ τ h(t) dq(t) = ⎣⎡ p(τ)⋅dτ⎦⎤⋅h(t − τ) q(t) • t because of the linearity of the system the cumulative response of the system to the t t – τ function p(t) is given by the superposition of all the infinitesimal responses t ⤷ q(t) = p(τ)h(t − τ)dτ ∫0 t NB p(τ) = net rainfall
h(t) is the INSTANTANEOUS UNIT HYDROGRAPH
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 11 Instantaneous Unit Hydrograph (IUH) - properties
• h(t) as “memory” or “weight” function ⤷ memory of q(t) for the input p(t), which occurred (t-τ) before ⤷ influence (“weight”) on q(t) due to the input p(t), which occurred (t-τ) before
• h(t) as probability density function ⤷ probability that a raindrop occurred at time t=0 in any place of the basin has to reach the outlet between t and t+dt
• h(t) is defined only in + ⤷ h(t) > 0 ∀t > 0
t t t+dt • H(t) à H (t) = h(t)dt ≤ 1; H (t) = h(t)dt ≤ h(t)dt = H (t + dt) ∫0 ∫0 ∫0 ⤷ S-curve = response to unit step input (constant intensity, infinite duration)
⤷ p, H(t)
t Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 12 Instantaneous Unit Hydrograph (IUH) – properties (2) p(t) h(t) t tR h(t) t h t p q(t) UH t
tp tL tUH t tH
• t = baselength of the hydrograph • tp = time to peak à mode of the pdf H • t = rainfall duration • hp = peak intensity à mode value R • t = IUH baselength • tL = time lag à mean of the pdf UH
tUH ⤷ tL = t ⋅h(t)dt = E ⎡h(t)⎤ ⤷ tH = tR +tUH ∫0 ⎣ ⎦
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 13 Discrete form of the IUH p(t) • pm = mean rainfall intensity in Δt
1 tm M=7 ⤷ pm = p(t)dt tm = mΔt m = 1,2,..., M Δt ∫tm−1 • discrete convolution integral k 1 2 3 4 5 6 7 t t 1 h(t) 1 unit ⤷ qk = ∑ pmΔH k−m+1 Δ = m=1
– ΔH = H t − H t n ( n ) ( n−1 ) N=6 – hydrograph baselength tH = k⋅Δt
– tN = IUH baselength, ΔHn=0 for n > N q(t) 1 2 3 4 5 6 t
– qk has k=M+N-1 values ≠ 0
example tH = 12 • q(4) = p(1)⋅h(4) + p(2)⋅h(3) + p(3)⋅h(2) + p(4)⋅h(1) = 24 units 1 2 3 4 5 6 7 8 9 1011 12 t Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 14 Example of IUH application (convolution integral) 1/2 t p(t) [h] [mm/h] 1 2 2 4 3 8 4 3 5 1 6 1 Q(t) t h(t) [h] [-] • A = 1 km2 1 1/20 • Δt = 1 h = 3600 s 2 3/20 3 3/10 t q = p ⋅h ⋅ Δt 4 1/4 • t ∑ m t−m+1 m=1 5 3/20 6 3/40 7 1/40 Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 15 Example of IUH application (convolution integral) 2/2 1 1 t = 1 à q1 = p1 ⋅h1 ⋅ Δt = 2 ⋅ ⋅1 = mm 20 10 • A = 1 km2 [mm/h]⋅[-]⋅[h] 1 1 1 • Δt = 1 h = 3600 s Q = q ⋅A⋅ = ⋅10−3 ⋅106 ⋅ = 0.0278 m3 /s 1 1 Δt 10 3.6 ⋅103 2 -1 t [mm]⋅[km ]⋅[s ] ⎡ 3 1 ⎤ 1 • qt = pm ⋅ht m 1 ⋅ Δt q p h p h t 2 4 1 mm ∑ − + t = 2 à 2 = [ 2 ⋅ 1 + 1 ⋅ 2 ]⋅ Δ = ⎢ ⋅ + ⋅ ⎥⋅ = m=1 ⎣ 20 20 ⎦ 2 1 1 −3 6 1 3 Q2 = q2 ⋅ A⋅ = ⋅10 ⋅10 ⋅ 3 = 0.139 m /s NB qt is computed per unit area Δt 2 3.6 ⋅10
à q = p ⋅h + p ⋅h + p ⋅h ⋅ Δt = ... t p(t) t h(t) t = 3 3 [ 3 1 2 2 1 3 ] 1 8 1 Q = q ⋅ A⋅ = ⋅10−3 ⋅106 ⋅ = 0.444 m3 /s [h] [mm/h] [h] [-] 3 3 Δt 5 3.6 ⋅103 1 2 1 1/20
2 4 2 3/20 t = 4 à … 3 8 3 3/10 t = 5 à … 4 3 4 1/4 … 5 1 5 3/20 6 1 6 3/40 7 1/40
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 16 IUH identification Two options • deconvolution à given observed q(t) and p(t) solve for h(t) the convolution integral t ⤷ q(t) = ∫ h(t − τ) p(τ)dτ à h(t) = … 0 • synthetic unit hydrographs (linear parametric) ⤷ empirical à generally characterised by prescribed shape and by functions of the time to peak and peak intensity h(t) e.g. triangular unit hydrograph t
⤷ conceptual à based on lumped parametric descriptions of the runoff concentration mechanisms e.g. basin storage and transfer represented by the hydraulic analogue of the linear reservoir P(t) h(t)
W(t) W(t) = k⋅Q(t) Q(t) t Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 17 Linear parametric IUHs
Hydraulic analogues provide a convenient framework
• linear reservoir à represents the storage and routing effects of the basin response through a linear dependence of the storage, W(t), from the outflow, Q(t)
W(t) = k⋅Q(t) P(t) h(t) where k is a storage coefficient representing the average delay imposed W(t) t by the reservoir type of basin response Q(t)
• linear channel à represents the basin response as kinematic transfer of the rainfall excess from any point in the watershed ⤷ the response is modulated (delayed) by the travel time from the place where the raindrop occurs and the watershed outlet à no routing (i.e. due to storage) effects
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 18 Linear reservoir IUH 1/2
Hypothesis: p(t) • synchronous transfer throughout the network à W = W[h(q)] ⇒ W(q)
• linear dependence of W on Q à W(t) = k⋅q(t) (•) dW (t) • mass continuity à p(t) − q(t) = (••) dt q(t)
(t−τ) ⎡ − ⎤ dq(t) t ⎢e k ⎥ (•) + (••) à k + q(t) = p(t) ⇒ q(t) = p(τ)dτ + q0 dt ∫0 ⎢ k ⎥ ⎣⎢ ⎦⎥
t h(t) 1 − ⤷ IUH à h(t) = e k à k where the parameter k 1/k is a storage constant
and tp = 0 ;hp = 1/k ;tL = k ;tUH ➞ ∞ t Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 19 Linear reservoir IUH 2/3 – hydrograph
h(t) constant rainfall intensity, h(t) constant rainfall intensity, infinite duration finite duration ϑ
t t
p(t) p(t)
p* p* t t ϑ q(t) q(t)
Qmax Qmax p* p* t t ϑ ϑ à Q = p * 1− e− k Qmax = p* is reached for t ∞ max ( ) − t rising limb à q(t) = p *(1− e k ) −t−ϑ − t falling limb à q(t) = p *(e k − e k ) Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 20 Linear reservoir IUH 3/3 – reservoir in series (Nash model 1/2) To better modulate the basin response through the storage effects n reservoirs of equal storage constant k can be used in a cascade
−t p(t) 1 k • for a unitary pulse à p(t)=1 à q1 (t) = e = I2 (t) k
outflow from the inflow to the q1 first reservoir second reservoir
• by applying the convolution t integral for the second linear à q2 (t) = I2 (t)h(t − τ) ∫0 q2 reservoir, one obtains t 1 −t 1 −(t−τ) = e k ⋅ e k dτ = ∫0 qn-1 k k t −t = e k k 2 qn n−1 1 ⎛ t ⎞ −t • by repeating for n reservoirs à h(t) = ⎜ ⎟ e k (n ∈ N) (n −1)!k ⎝ k ⎠
α−1 1 ⎛ t ⎞ −t ∞ ∈ à ➞ à k α−1 − x for (n + ) n α h(t) = ⎜ ⎟ e where Γ(α) = x e dx is the Gamma function Γ(α)k ⎝ k ⎠ ∫0 Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 21 Linear reservoir IUH 3/3 – reservoir in series (Nash model 2/2)
The characteristics of the Nash model depend on the value of the parameters, n (or α) and k
ê
h(t integer # of non-integer # of ) n = 1 reservoirs n reservoirs, α
n = 2 k = 1 tL = n⋅k tL = α⋅k n = 5 t = (n-1) ⋅k t = (α-1) ⋅k p p n = 10 n = 15 n−1 α−1 (n −1) −(n−1) (α −1) −(α−1) h = e hp = e p k(n −1)! kΓ(α) t
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 22 Linear channel IUH – travel time concept
• the time required to a water particle to travel the distance L from A to B depends on its velocity, v(l) L ⤷ dl = v(l)⋅dt à t = dl v(l) ∫0
⤷ if v(l) = vi = constant for Δli distance increments, i=1, …, I I ⤷ t = Δl v ∑i=1 i i
• tc, time of concentration is the time required to a water particle to travel from the farthest point of the watershed to the outlet à the time at which all of the watershed begins to contribute
• tc can be estimated ⤷ “directly” ⤷ through empirical equations, field measurements or assuming the isochrones to coincide with contour lines ISOCHRONES : ⤷ indirectly ⤷ through the knowledge of the network topology and the lines of equal estimation of velocity TRAVEL TIME to à from channel geometry and channel flow equations the outlet à field measurements à tables Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 23 Linear channel IUH – time of concentration emp. equations (1/2)
tc = f(basin morphology and characteristics) ê pay attention to range/conditions of validity
[Chow et al. 1998, p. 500] Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 24 Linear channel IUH – time of concentration emp. equations (2/2)
tc = f(basin morphology and characteristics)
ê pay attention to range/conditions of validity
[Chow et al. 1998, p. 501]
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 25 Linear channel IUH – velocity estimation
[Chow et al. 1998, p. 165] Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 26 Linear channel IUH (time-area method) Hypotheses: • flow moves as liquid mass transfer • flow particles move independently from each other • flow movement depends on the position in the catchment ⬇︎ • for a rainfall intensity i(t) and a contributing area dA the travel time to resulting flow dq(t) is à dq(t)= i(t-τ)⋅dA outlet * outlet from • because of the linearity of the system à superposition of the area = A dA effects *
A A ⤷ the resulting flow at the outlet is due to the areas contributing each with a travel time dictated by its position ⬇︎ area, t A(t) • assuming rainfall unitary pulses à h(t) = δ(t − τ)dA tc ∫0 h(t) • assuming isochrones corresponding to contour lines ⤷ t dA h(t) = δ(t − τ) dτ IUH ∫0 dτ • the IUH corresponds to the derivative of the time-area curve t Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 27 Linear channel IUH (time-area method) example TIME-AREA CURVE
TOTAL AREA A* t compute the • hydrograph from a i(t) constant, ∞ duration rainfall input i(t)
• hydrograph from a t variable, finite duration rainfall input
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 28 Linear channel IUH (time-area method) example
t
convolution k q(kΔt) = ijΔt ⋅ Ak− j+1 integral ∑ j=1 i(t)
for i(t) = i = constant q(t) * Qmax = i A if t < tc à q(t) = i⋅A(t)
t > t à q(t) = i⋅A* if c t
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 29 Linear channel IUH (time-area method) example
i(t)
t convolution k q kΔt = i ⋅ A integral ( ) ∑ j=1 jΔt k− j+1 time step discharge q(t) Δt q(Δt) = i1 ⋅ A1 2Δt q(2Δt) = i2 ⋅ A1 + i1 ⋅ A2 3Δt q(3Δt) = i3 ⋅ A1 + i2 ⋅ A2 + i1 ⋅ A3 … … t c qmax t Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 30 Triangular IUH
The shape is based on the emprical observation of flood hydrographs. The IUH is determined through
• the time to peak, tp • the peak intensity, h p h(t) ⤷ the resulting IUH is ⤵︎
⎧hp ⋅t t p 0 ≤ t ≤ t p ⎪ hp ⎪ hptUH hpt h(t) = ⎨ − t p ≤ t ≤ tUH t − t t − t ⎪ UH p UH p t ⎪0 t > t ⎩ UH t t ⤷ and the S-curve, H(t) is ⤵︎ p UH
2 H(t) ⎧ hp 2t p t 0 ≤ t ≤ t p ⎪( ) 1 ⎪ 1 hp 2 2 t H (t) = ⎨− t − t − t p ≤ t ≤ tUH ⎪ 2 tUH − t p tUH − t p tUH − t p inflection point ⎪1 t > t ⎩ UH t
• where the UH baselength is tUH=2/hp and the time lag is tL=1/3(tp+tUH)
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 31 Mixtures of conceptual IUHs
The linear reservoir and linear channel conceptual IUHs can be used to build more complex models, which aim at better representing the complexity of the response, e.g.:
• 2 linear reservoirs of different storage constant to represent the fast surface runoff and the slow subsurface flow
• Clark’s model à the response of the watershed is described by a combination of linear channel and linear reservoir method
e−(t−τ) k dA(τ) ⤷ h(t) = dτ ∫ k dt
• the watershed is divided in sub-watersheds the response of which is modelled by a linear reservoir which is drained by a linear channel
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 32 Linear R-R models – parameter estimation (1/2)
observed data R-R models require the estimation of RAINFALL or the parameters of the infiltration input DDFs + synthetic and of the runoff concentration hyetograph model components
e.g. SCS-CN INFILTRATION model parameters: CN, α
R model RUNOFF -
R CONCENTRATION ê Nash IUH parameters: n, k basin response function Parameter estimation consists of q(t) observed computed tuning the values of the parameters to achieve a match between
output DISCHARGE computed and observed hydrologic variables (typically the discharge) t Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 33 Linear R-R models – parameter estimation (2/2)
• Parameter estimation à matching of observed and computed values q(t) observed • Computed values are function of model equations, which are function computed of parameters, e.g.
⤷ qcomputed(t) = f(infiltration + runoff concentration parameters) Parameter estimation can be carried out by: • manual methods t ⤷ trial and error (iterative): parameters are adjusted manually until a convergence of computed and observed values is reached (visual check vs numerical metrics) ⤷ method of moments (non-iterative): matching of the sample moments (computed from observations) with moment theoretical expressions (function of parameters) • automatic methods ⤷ least squares à minimisation of an objective function, F, based on one or more goodness of fit criteria à e.g. average of the square error between observed and computed variable (e.g. flow):
1 N 2 ⤷ ε2 = q − q à F = min ε2 àconvergence of obs. and comp. values N ∑i=1( obs comp ) { } NB1: iterative methods require to define a criterion of convergence à goodness of fit measures NB2: parameter values should always have plausible values
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 34 Nash model parameter estimation by method of moments (1/2)
Hyetograph, UH and hydrograph can be characterised by their t moments: I j m max I t − t * ∑ j=1 j ( j I ) • hyetograph, moment order m à M = Im jmax i(t) I j h(t) ∑ j=1 j m h max h t − t * ∑ j=1 j ( j h ) • UH, moment order m à M h = j m max h t ∑ j=1 j
j m max Q t − t * q(t) ∑ j=1 j ( j Q ) • hydrograph, moment order m à M Q = j m max Q ∑ j=1 j
Q * * * where are the coordinates of the center of mass of hyetograph, UH tI , th , tQ and hydrograph t
* For a linear system it holds: M = M − M t hm Qm Im Q
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 35 Nash model parameter estimation by method of moments (2/2)
• The UH moments are function of the UH parameters, n and k
t • For the linear reservoir UH it is sufficient to compute the moment I of 1st order (one unknown parameter à one moment equation):
⤷ k M M = Q1 − I1 i(t) h(t) • For the Nash UH it is necessary to compute the moment of 1st and h 2nd order (two unknown parameter à two moment equation):
t ⤷ ; M = k ⋅n = t M k 2 n k t h1 L h2 = ⋅ = ⋅ L q(t) M M M which, combined with hm = Qm − Im , allow to estimate the Nash model parameters from the system of equations: Q k ⋅n = M Q − M I 1 1 (•) k 2 ⋅n = M − M t Q2 I2
* • k and n can be estimated by substituting into (•) the moments t Q Mˆ , Mˆ , Mˆ , Mˆ computed from observed concurrent I 1 I 2 Q 1 Q 2 hyetographs and hydrographs Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 36 Rainfall-runoff modelling
Engineering Problems: ⤷ Design flood estimation for flood protection measures ⤷ Design of urban drainage systems Solution ⤷ Design Flood Approach (peak, volume and duration estimation for a given RP)
Method ⤷ DDF + synthetic hyetograph ⤷ Infiltration model ⤷ Runoff concentration model (IUH)
Hydrology – Rainfall-Runoff Transformation – Autumn Semester 2017 37