1European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

Salvatore Manfreda1,2 and Ignacio Rodríguez-Iturbe1

1 Princeton University 2 Università degli Studi della Basilicata

Space-Time Soil Moisture Dynamics: Stochastic Structure and Sampling

Requirements

European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

2European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

Space-Time Soil Moisture Dynamics: Outline

Soil moisture dynamics driven by stochastic rainfall;

Effects of averaging the soil moisture in space and time;

Effects of vegetation heterogeneity on soil moisture dynamics;

Sampling of soil moisture fields using random or random stratified sampling;

Effects of spatial heterogeneity of vegetation on soil moisture sampling.

3European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

Model performance versus model complexity

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Rainfall model

(Cox & Isham, 1988)

Rainfall occurrences are modeled by a sequence of circular rain cells that occur in a Poisson process of rate λR in space and time.

Each cell is characterized by a random radius, WR, and also random duration and intensity.

μD mean value of rainstorm duration (η=1/μD).μR mean cell radius (ρ=1/μR).μX mean rainfall intensity (β=1/μX).

Parameters:

5European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

Schematic representation of the Soil Water Balance

S(t) [-] relative soil moisture at time t; Y(t) [L/T] rainfall forcing; n [-] soil porosity; Zr [L] active soil depth; L(s) [L/T] leakage function of s;E(s) [L/T] evapotranspiration function of s;(1-Φ) [-] net rainfall coefficient.

Zr

(parameter: n, Zr e V)

Rainfall (parameters: λR, ρR, η e β)

Evapotranspiration(parameter: V)

Leakage(parameter: V)

Interception(parameter: Φ)

Net Precipitation

L(s) + E(s) = V S(t)

V [L/T] water loss coefficient.

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Standardization of the soil moisture balance equation in space

Soil moisture dynamics in equilibrium

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Theoretical covariance function of the relative soil moisture

0 0.2 0.4 0.6 0.8 10

0.02

0.04

0.06

0.08

0.1

0.12

0.14=0.0

=0.1

=0.2

=0.3

=0.4

<Y>/V [-]

S [

-]

Hyperarid

Arid Semi-aridDry subhumid Humidb2/(aη) = (1-Φ)2 /(nZr V η).

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Soil moisture correlation with homogeneous vegetation across the landscape

1020

3040

50

1020

3040

50

0

0.5

1

Time [day]

Correlation of soil saturation

distance [km]

1020

3040

50

1020

3040

50

0

0.5

1

Time [day]

Correlation of soil saturation

distance [km]

nZr = 100 mm nZr = 400 mm

…using rainfall parameters estimated over Southern Italy.

Increasing the effective Soil depth

Increases the correlation

in time

2

41,;,0

lR

hahR

el

a

aeehtlStScorr

9European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

The averaged processThe variance of the relative soil moisture process averaged over a given square area A of side L can be obtained integrating the covariance of the soil moisture process in space

(e.g., Vanmarcke, 1983)

10European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

Standard Deviation of the Averaged Soil Moisture in Space and Time

obtained using the Gaussian approximation to the spatial correlation function

Comparison of analytical approximation with numerical integration

222

3322 112

Taa

TeaaTe TaT

SS LTL

The variance of the process averaged over a square spatial region of side L and a temporal interval of length T

The variance of the instantaneous soil saturation process averaged over a square area of size L × L

11European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

Heterogeneous Vegetation Savanna (Australia)

Savanna (Africa)

Biomes (L. Daniels, 2004)

Tropical Savanna (Bolivia)

12European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

Vegetation PatternThe landscape is given by the combination of two functionally different vegetation types (e.g., grasses and trees).

Trees are located according to a Poisson process in space with rate λT and have circular crowns with radii, RT, exponentially distributed with parameter ρT.

LandscapeGrass

Trees

RT

13European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

Specification vegetation model

Landscape

Grass

Trees

RTAc

Bu

l

The probability that a point is covered by a tree is

The probabilities of the four different possible combinations of vegetation cover at two points A and B separated by a distance l in space are as follows

where

14European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

The soil moisture correlation function changing the landscape

λT = 500 km-2 ρT

-1 = 8 m

Tree cover = 0.18%

λT = 5000 km-2 ρT

-1 = 8 m

Tree cover = 85%

λT = 1500 km-2 ρT

-1 = 8 m

Tree cover = 45%

Rainfall forcingVegetation heterogeneity

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Effects of Spatial Heterogeneity on the Soil Moisture Variability

R is the empirical autocorrelation estimated from soil moisture fields in Illinois (Vinnikov, 1999).

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On the Spatial and Temporal Sampling of Soil Moisture Fields

The long term mean soil moisture for a given time interval during a given season (e.g., daily soil moisture during month of June) at any point of a statistically homogeneous region (mS).

The mean soil moisture over an area SA,

where S(xi) is the soil moisture at a site xi and represents a realization of the soil moisture process over a region assumed statistically homogeneous.

17European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

The long term mean soil moistureAssuming N sample points in space operating during T days of the same statistically homogeneous season, mS is estimated through S as given by,

The goodness of the estimation is measured through the variance of S,

18European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

The long term mean soil moistureFollowing Rodriguez-Iturbe and Mejia (1974), the variance of S can be written as

The above equation may be written as the product of two reduction factors affecting the variance of the daily soil moisture at a point

Variance reduction factor due to the temporal sampling

Variance reduction factor due to the spatial sampling

where

19European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

Spatial and temporal variance reduction factorThe time-dependent factor, F1(T), is given by

The space-dependent factor, F2(N), is

where r(t) represents the correlation function in time and r(x) in space (Rodriguez-Iturbe and Mejia, 1974).

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Variance reduction factor due to the spatial sampling (with uniform vegetation)

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Variance reduction factor due to the spatial sampling (with heterogeneous vegetation)

Vegetation cover

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Mean soil moisture over an area for any given dayPerformance of a network with random design

Performance of a network with random stratified design

(Rodrìguez-Iturbe & Mejìa, 1974)

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The sampling of daily soil moisture (homogeneous vegetation cover)

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The sampling of instantaneous soil moisture (heterogeneous case)

Vegetation cover

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Nature is Complex, butsimple models may suffice. (J. Sprott)

Definition of a feasible mathematical characterization in space and time of soil moisture dynamics (in arid or semi-arid environment);

Definition of the effects of time-space averaging of the relative soil moisture process;

Description of the effects of vegetation heterogeneity on soil moisture dynamics;

Quantitative estimate of the sampling errors within a soil moisture network.

1. The spatial geometry has a significant impact on the sampling of the SA, while it is less relevant for mS.

2. In the case of mS, the length of the record is a commanding factor in what concerns the variance of estimation, specially for soils with shallow rooted vegetation.

3. Spatial vegetation heterogeneity plays an important role on the variance of estimation of the soil moisture, being particularly critical for the sampling of SA.

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Future directions

Water Budget at the Basin Scale;

Interaction between Soil Water and Vegetation Dynamics;

Soil Moisture and Nitrogen Interactions.

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Future directions

Vegetation dynamics and soil water interaction.

0 1000 2000 3000 4000 50000

50

100

Rai

n (m

m/d

ay)

0 1000 2000 3000 4000 50000

20

40

Ass

imil

atio

n (g

/m2 )

0 1000 2000 3000 4000 50000

0.5

1

s (

/n)

0 1000 2000 3000 4000 50000

1

2

Bio

mas

s (k

g/m

2 )

time (days)

Stochastic rainfall forcing…

Constant rainfall…

0 1000 2000 3000 4000 50000

0.5

1

s (

/n)

0 1000 2000 3000 4000 50000

0.5

1

Bio

mas

s (k

g/m

2 )

time (days)

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Acknowledgments

NOOA under the grant #NA17RJ2612.

NSF under the National Center for Earth Surface Dynamics (NCED).

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Papers related to this line of work … Manfreda, S. &  I. Rodríguez-Iturbe, Space-Time Soil Moisture Dynamics:

Stochastic Structure and Sampling Requirements, Advances in Water Resources (in preparation), 2005.

Manfreda, S. &  I. Rodríguez-Iturbe, On the Spatial and Temporal Sampling of Soil Moisture Fields, Water Resources Research (in press), 2006.

Rodríguez-Iturbe, I., V. Isham, D.R. Cox, S. Manfreda, A. Porporato, Space-time modeling of soil moisture: stochastic rainfall forcing with heterogeneous vegetation, Water Resources Research, VOL. 42, W06D05, doi:10.1029/2005WR004497, 2006.

Isham, V., D.R. Cox, I. Rodríguez-Iturbe, A. Porporato, S. Manfreda, Representation of Space-Time Variability of Soil Moisture. Proceedings of the Royal Society A: Mathematical, Physical and Engineering Sciences, 461(2064), 4035 – 4055, (doi:10.1098/rspa.2005.1568), 2005.

30European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.

Thanks for you attention…