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- 1. Space-Time Soil Moisture Dynamics: Stochastic Structure and Sampling Requirements Salvatore Manfreda1,2 and Ignacio Rodríguez-Iturbe1 Princeton University 1 2 Università degli Studi della Basilicata 1 European Geosciences Union, General Assembly 2006, Vienna, Austria,Vienna,April 2006. – 07 April 2006. European Geosciences Union, General Assembly 2006, 02 – 07 Austria, 02
- 2. 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. European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 2
- 3. Model performance versus model complexity European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 3
- 4. Rainfall model • 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. Parameters: μD mean value of rainstorm duration (η=1/μD). μR mean cell radius (ρ=1/μR). μX mean rainfall intensity (β=1/μX). (Cox & Isham, 1988) European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 4
- 5. Schematic representation of the Soil Water Balance Rainfall (parameters: λR, ρR, η e β) S(t) [-] relative soil moisture at time t; Evapotranspiration Y(t) [L/T] rainfall forcing; (parameter: V) n [-] soil porosity; Zr [L] active soil depth; Interception L(s) [L/T] leakage function of s; (parameter: Φ) E(s) [L/T] evapotranspiration function of s; (1-Φ) [-] net rainfall coefficient. Net Precipitation L(s) + E(s) = V S(t) Zr (parameter: n, Zr e V) Leakage V [L/T] water loss coefficient. (parameter: V) European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 5
- 6. Standardization of the soil moisture balance equation in space Soil moisture dynamics in equilibrium European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 6
- 7. Theoretical covariance function of the relative soil moisture 0.14 φ=0.0 0.12 φ=0.1 Hyperarid φ=0.2 0.1 φ=0.3 0.08 φ=0.4 σ [-] S 0.06 0.04 0.02 Dry b /(aη) = (1-Φ) /(nZr V η). 2 2 0 Arid Semi-arid subhumid Humid 0 0.2 0.4 0.6 0.8 1 <Y>/V [-] European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 7
- 8. Soil moisture correlation with homogeneous vegetation across the landscape corr ( S ( 0, t ) ; S ( l , t + h ) ) = (ηe − ah − ae −ηh ) 1 + ρ l e R −ρR l 2 (η − a ) 4 Correlation of soil saturation Correlation of soil saturation Increasing the effective Soil depth 1 1 0.5 Increases 0.5 the correlation 0 10 in time 0 10 10 20 10 20 20 30 20 30 30 40 Time [day] 30 40 Time [day] distance [km] 40 40 50 50 distance [km] 50 50 nZr = 100 mm nZr = 400 mm …using rainfall parameters estimated over Southern Italy. European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 8
- 9. The averaged process The 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) European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 9
- 10. Standard Deviation of the Averaged Soil Moisture in Space and Time The variance of the instantaneous soil saturation process averaged over a square area of size L × L Comparison of analytical approximation obtained using the Gaussian approximation to the with numerical integration spatial correlation function The variance of the process averaged over a square spatial region of side L and a temporal interval of length T σ 2 =σ 2 ( ( ) ( 2 η 3 e − aT + aT − 1 − a 3 e −ηT + ηT − 1 )) a 2η 2 (η − a )T 2 T SL SL European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 10
- 11. Heterogeneous Vegetation Savanna (Australia) Savanna (Africa) Tropical Savanna (Bolivia) 11 European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. (L. Daniels, 2004) Biomes
- 12. Vegetation Pattern The 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. Landscape Grass RT Trees European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 12
- 13. Specification vegetation model 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 Grass where Landscape RT Ac l Bu Trees European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 13
- 14. The soil moisture correlation function changing the landscape λT = 500 km-2 ρT-1 = 8 m Tree cover = 0.18% λT = 1500 km-2 ρT-1 = 8 m Tree cover = 45% λT = 5000 km-2 ρT-1 = 8 m Tree cover = 85% Vegetation Rainfall forcing heterogeneity European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 14
- 15. Effects of Spatial Heterogeneity on the Soil Moisture Variability R is the empirical autocorrelation estimated from soil moisture fields in Illinois (Vinnikov, 1999). European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 15
- 16. 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. European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 16
- 17. The long term mean soil moisture Assuming 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, European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 17
- 18. The long term mean soil moisture Following Rodriguez-Iturbe and Mejia (1974), the variance of S can be written as where 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 Variance reduction factor due to the temporal sampling to the spatial sampling European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 18
- 19. Spatial and temporal variance reduction factor The 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). European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 19
- 20. Variance reduction factor due to the spatial sampling (with uniform vegetation) European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 20
- 21. Variance reduction factor due to the spatial sampling (with heterogeneous vegetation) Vegetation cover European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 21
- 22. Mean soil moisture over an area for any given day Performance of a network with random design Performance of a network with random stratified design (Rodrìguez-Iturbe & Mejìa, 1974) European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 22
- 23. The sampling of daily soil moisture (homogeneous vegetation cover) European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 23
- 24. The sampling of instantaneous soil moisture (heterogeneous case) Vegetation cover European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 24
- 25. Nature is Complex, but simple 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. • The spatial geometry has a significant impact on the sampling of the SA, while it is less relevant for mS. • 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. • Spatial vegetation heterogeneity plays an important role on the variance of estimation of the soil moisture, being particularly critical for the sampling of SA. European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 25
- 26. Future directions Water Budget at the Basin Scale; Interaction between Soil Water and Vegetation Dynamics; Soil Moisture and Nitrogen Interactions. European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 26
- 27. Future directions Stochastic rainfall forcing… Vegetation dynamics 100 Rain (mm/day) and soil water 50 interaction. 0 0 1000 2000 3000 4000 5000 1 s (θ/n) 0.5 Constant rainfall… 0 1 0 1000 2000 3000 4000 5000 40 Assimilation (g/m ) 2 /n) 0.5 s (θ 20 0 0 1000 2000 3000 4000 5000 0 0 1000 2000 3000 4000 5000 1 2 Biomass (kg/m ) Biomass (kg/m ) 2 2 0.5 1 0 0 0 1000 2000 3000 4000 5000 0 1000 2000 3000 4000 5000 time (days) time (days) European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 27
- 28. Acknowledgments NOOA under the grant #NA17RJ2612. NSF under the National Center for Earth Surface Dynamics (NCED). European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 28
- 29. 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. European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 29
- 30. Thanks for you attention… European Geosciences Union, General Assembly 2006, Vienna, Austria, 02 – 07 April 2006. 30

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