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