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Consistent Determination of Antecedent Soil Moisture (Retention Parameter) of the SCS
Method Based on Stochastic Soil Water Balance
Mt Washington, NH
grants NFS-CBET-1033467, NSF-EAR-1316258, CNPq-402871/2012-0 (PVE)
SWAT Conference: Hydrology 2014, Porto de Galinhas, July 30-Aug 1, 2014
Mark Bartlett1, Suzana Montenegro2, Antonio Antonino2, and Amilcare Porporato1
1Duke University, NC USA2Universitade Federal de Pernambuco (UFPE), PE Brazil
Motivation and Overview
• The SCS curve number is a widely used (including in the SWAT model) empirical method for runoff quantification
• Based on extensive observations, it provides robust estimates of runoff at the event scale
• However, the antecedent soil moisture conditions (retention parameter) are linked to soil, plant and climatic conditions through empirical tables and charts having little physical basis
• The stochastic soil moisture models used in ecohydrology (e.g., Porporatoet al. AWR, 2002; Rodriguez-Iturbe and Porporato, Ecohydrology, Cambridge University Press, 2004) provide probabilistically consistent determination of antecedent soil moisture conditions based on soil, plant and climate characteristics but with rather crude runoff representation
Combining the theoretical framework of stochastic ecohydrologymethods with the empirical strength of the SCS method
After the Dust Bowl of the ‘30s, the US Congress declares soil erosion “a national menace”, establishing the Soil Conservation Service (USDA) to develop extensive conservation programs that retain topsoil and prevent irreparable damage to the land.
SCS Curve Number Method: Background
Buried machinery in barn lot, Dallas, South Dakota, May 1936
Dust storm approaching Stratford, Texas, April 18 1935
The issue is still very actual, multidisciplinary and complex, involving ecohydrology, geomorphology, plant physiology, biogeochemistry, etc.
CalhounCritical Zone Observatory
Critical Zone Observatories
Calhoun, 1950 circa
• The SCS curve number equation for runoff is
SCS Curve Number
𝑄 =𝑅 − 𝐼
𝑅 − 𝐼 + 𝐵
where:Q = depth of runoffR = depth of rainfallI = initial abstractionB = basin ret. par.
𝐶𝑁 =1000
10 +𝐵 (mm)25.4
Part 630 Hydrology, National Engineering Handbook
S(t)
B(t)
R-I
Q
0.05
0.15
0.25
150 200 250 300 350
Julian Day
q (
%)
interspace
canopy
0
5
10
15
20
Pre
cip
ita
tio
n
(mm
day
)
Lumped parameterization of spatial response of watershed due to different soil moisture conditions
s
Curve number data
Curve number selection
Tables for antecedent moisture conditions
10
Stochastic soil water balance used in ecohydrology
Runoff
INPUT: RAINFALL
(intermittent-
stochastic)
t
h
Evapo-
transpiration
Troughfall
Zr
Effective porosity, n
Zr
Effective porosity, n
Leakage
Runoff
n = porosity
Zr = active soil depth
s(t) = relative soil moisture
R(t) = rainfall rate
I(t) = canopy interception
Q[s(t),t] = runoff
E[s(t),t] = evapotranspiration
L[s(t),t] = leakage
)()(),()()()(
tsLtsEttsQtItRdt
tdsnZr
Transpiration
0 20 40 60 80 100
1
2
3
4
5
6
7
Adapted from: Gollan et al., Oecologia,
65, 356-362, 1985.
After: Federer, Water Resour. Res., 15
(3), 555-561, 1979.
Extractable soil water
Sto
mat
al c
on
du
ctan
ce
stomata
xylem
vessels
soil-root
interface
Losses
1
Leakage
Emax
Evapotranspiration
Ks
50 100 150 200tHdL
0.2
0.4
0.6
0.8
1
s
sw
s*
sfcsw s* s
1-D Non-linear (stochastic) differential equation
driven by a state-dependent Poisson noise
)()(),()()()(
tsLtsEttsQtItRdt
tdsnZr
Nonlinear lossesRnd jump input
s
us duetuptspstsps
tspt
0
,,,,
Probabilistic description: Chapman-Kolmogorov Eq.
jumps
0decaydet.
d)(d);(),()(d),()1(, suuuusptuuptsstssptttsp
s
y
u
s
Rainfall event,
prob t b(s-u; u)
t
Deterministic decay,
prob (1-)t
)(sdt
ds
su
dusExp
s
csp
)()()(
Steady-state PDF of soil moisture
Rodriguez-Iturbe et al., Proc.
Royal Soc. A, 455, 3789, 1999
Porporato et al. Am. Nat., 2004
Daly and Porporato PRE 2010
Salvucci G. (2001) Water Resour. Res. 37(5), 1357-1365.
Experimental verification
Jump (infiltration) distribution for SCS method
𝑦|𝑏 =𝑒𝛽𝛾𝑏 − 1 + 𝛾𝑏 + 𝑏2𝑒𝛾𝑏𝛾2 Chi 𝛾𝑏 + Shi 𝛾𝑏
𝛾𝑒𝛽𝛾𝑏
𝑝𝑧 𝑧 = 𝛾𝑒−𝛾𝑧
For an exponential distribution of rainfall, i.e.,
The average infiltration is
Approximate infiltration with a new exponential distribution, i.e.,
𝑝𝑦 𝑦|𝑏 =1
𝑦|𝑏𝑒−
1
𝑦|𝑏𝑦
b=0.8
b=0.2
𝑝𝑦 𝑦|𝑏 =
𝑝𝑧 𝑦 𝑦 ≤ 𝛽𝑏
𝑏2
𝑏−(𝑦−𝛽𝑏) 2𝑝𝑧 𝛽𝑏 +
𝑏(𝑦−𝛽𝑏)
𝑏−(𝑦−𝛽𝑏)𝑦 > 𝛽𝑏
PDF of retention parameter
𝑝𝑏(𝑏) = 𝐶𝑦|𝑏 −𝜆 𝑘
Γ 𝜆 𝑘𝑒− 1−𝑏 𝑦|𝑏 1 − 𝑏
𝜆𝑘−1
α=1.5 cmw0=24 cm (max. basin retention)
PDF of curve number (CN)
λ=0.15 d-1
λ=0.45 d-1
λ=0.15 d-1
λ=0.45 d-1
λ=0.3 d-1
λ=0.3 d-1𝐶𝑁 =1000
10 +𝐵 (mm)25.4
Individual realizations
𝑅
Rainfall PDF 𝒑𝑹(𝑹)
Ru
no
ffP
DF 𝒑𝑸(𝑸)
𝑅
Rainfall PDF 𝒑𝑹(𝑹)
Ru
no
ffP
DF 𝒑𝑸(𝑸)
wet (Higher freq. of rainfall)
wet (Higher freq. of rainfall)
Budyko-type curve[1] for averagehydrologic balance
𝐷𝑟𝑦𝑛𝑒𝑠𝑠 𝐼𝑛𝑑𝑒𝑥,𝑫𝑰 =𝑃𝑜𝑡𝑒𝑛𝑡𝑖𝑎𝑙 𝑙𝑜𝑠𝑠 = 𝜂
𝐴𝑣𝑔. 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 = 𝜆𝛼
𝑬
𝑹=𝐴𝑣𝑒𝑟𝑎𝑔𝑒 𝑐𝑜𝑛𝑡. 𝑙𝑜𝑠𝑠 = 1 − 𝑏 𝜂
𝐴𝑣𝑔. 𝑟𝑎𝑖𝑛𝑓𝑎𝑙𝑙 = 𝜆𝛼
Average continuous losses
Average runoff losses
[1] Rodriguez-Iturbe, I. and Porporato, A. (2004). Ecohydrology of Water-Controlled Ecosystems, p. 55-56.
𝜆 Average frequency of input events 𝛼 Average input amount (normalized)𝜂 is the normalized max. continuous loss rate
𝑬
𝑹
less basin storage more runoff losses𝑬
𝑹min CN=50
min CN=77
min CN=89
min CN=97
Conclusions• We combined: - stochastic methods of soil water balance
- Empirical runoff formulation of SCS – CN method used by the SWAT model
• Result: a parsimonious but realistic representation of the water balance, which improves ecohydrological models and determines consistently the retention parameter
• This allows long term analyses of the effects of changes in PET, freq. of rainfall, etc. on the soil water balance
• Easily coupled to other processes (water stress, biogeochemistry) to perform probabilistic risk analysis and uncertainty quantification when planning for sustainable use of soil and water resources
grants NFS-CBET-1033467, NSF-EAR-1316258, CNPq-402871/2012-0 (PVE)
Conceptual probabilistic framework
Inputs (random pulses) at freq. λI with avg.
quantity αI
Runoff (pulses) partitioned by the SCS curve
number
𝑤0𝑑𝑠(𝑡)
𝑑𝑡= 𝐼 𝑡 − 𝑓𝐿 𝑠 𝑡 − 𝑄 𝑠(𝑡)
Rate for continuous water loss
Totalstoragedepth
Probabilistic description of the storage of a simple reservoir component
𝜕
𝜕𝑡𝑝𝑏 𝑏; 𝑡 =
𝜕
𝜕𝑏𝑓𝐿 𝑏 𝑝𝑏 𝑏; 𝑡 − 𝜆𝑝𝑏 𝑏; 𝑡 + 𝜆
𝑠
𝑝𝑧 𝑢 − 𝑏 𝑝𝑠 𝑢; 𝑡 𝑑𝑢
+𝜆 𝑢2
𝑢 − 𝑢 − 𝑏 − 𝛽𝑢2 𝑝𝑧 𝛽𝑢 +
𝑢 𝑢 − 𝑏 − 𝛽𝑢
𝑢 − 𝑢 − 𝑏 − 𝛽𝑢𝑝𝑏 𝑏; 𝑡 𝑑𝑢
Infiltration before initial abstraction
Continuous loss term 𝑓𝐿 𝑠 𝑡
Infiltration after initial abstraction
Probability density function (PDF) of normalized basin retention, b
Loss from the jump