space-time soil moisture dynamics
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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.
6European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.
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).
8European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.
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
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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).
16European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.
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).
20European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.
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.
27European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.
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)
28European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.
Acknowledgments
NOOA under the grant #NA17RJ2612.
NSF under the National Center for Earth Surface Dynamics (NCED).
29European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006.
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…
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