Space-time models for soil moisture dynamics

1. Space-time models for soil moisture dynamics Valerie Isham Department of Statistical ScienceUniversity College Londonvalerie@stats.ucl.ac.uk, http://www.ucl.ac.uk/stats/

2. Collaborators:David CoxNuffield College, OxfordIgnacio Rodriguez-Iturbe Civil and Environmental Engineering, PrincetonAmilcare Porporato Civil and Environmental Engineering, Duke

3. Overview Introduction Temporal models of soil moisture at a single-site point rainfall (ie concentrated at discrete time points) Spatial-temporal models of soil moisture spatially-distributedrainfall (at a point or temporally distributed in time) variable vegetation properties at a point and averaged over space-time Coupled dynamics of biomass and soil moisture temporal process at a single site Summary and future directions

4. Fundamental problem of hydrological interest… Soil moisture (and its spatial and temporal variability) is the dynamic link between climate, soil and vegetation, and impacts processes at a range of spatial scales. Point scale: infiltration, plant dynamics, biogeochemical cycle Hillslope: controlling factor for slope instability and land slides Basin: drought assessment, flood forecasting Region/continent: interaction with atmospheric phenomena

5. Soil moisture…… • increases due to precipitation • decreases due to evapotranspiration • and leakage • and is dependent on • soil properties • vegetation

6. We consider dynamics • at a daily time scale (no effects of diurnal fluctuations in temperature on evapotranspiration) • within a single season • on relatively small spatial scales (no feedback between soil moisture and rainfall). • The impact on the vegetation as well as of the vegetation is of interest. La Copita, Texas; courtesy of Amilcare Porporato/ Steve Archer

7. 20 15 ) 10 Precipitation (mm day 5 0 Precipitation and soil moisture Sevilleta, New Mexico courtesy of Amilcare Porporato/Eric Small

8. Temporal process of soil moisture • Modelling approach • We use • piecewise deterministic Markov processes (Davis 1984) in continuous time: sample paths have • periods of deterministic change governed by a differential equation • random jumps occurring at random times

9. Times: a Poisson process, rate Jumps: iid, density g Decay: constant rate S(t) X2 X3 X4 X1 T1 T2 T3 T4 t A very simple such process …… S(t): the Takács virtual waiting time process for a M/G/1 queue ie the service requirement of all the customers in the system at t, Alternatively:S(t) is the content of a store (reservoir) * replenished by random amounts at random times * subject to depletion at a constant rate when non-empty

10. Let and let Shave density for s> 0 Forward equation: Many properties of the process can be determined Special case: Xi~ exp( ) Equilibrium: if

11. Other properties and extensions (Cox and Isham, 1986) • transient solution: Laplace transform (wrt to t) of the moment generating function • expansions determining convergence to equilibrium • autocovariance function, in equilibrium • slowly varying arrival rate • (small )

12. For soil moisture • state-dependent decay • losses depend on current soil moisture level • boundedness of soil moisture • excess rainfall runs off saturated soil • state-dependent jumps, density g(x,s)

13. Soil moisture balance equation: n soil porosity Zrdepth of root zone I(random) rate of infiltration (dependent on ground cover) Erate of evapotranspiration (dependent on vegetation) L rate of leakage (dependent on soil properties) StandardiseI, E, L

14. Losses…approximated by 0 s* s11.0

15. distribution of infiltration…Assume that standardised infiltration I*(s,t) has an exponential ( ) distribution, truncated at 1- s The excess rainfall is lost as surface run-off.

16. Forward equation… for density of S(t) (no atom at origin since ). Equilibrium distribution (Rodriguez-Iturbe et al 1999) has the form Use piecewise linear form of continuity of p(s) at s* and s1. Normalise to 1 to find c.

17. Note: the atom of probability at 1-sin the state dependent jumps is not used explicitly in the derivation. Soil saturation only affects the restricted range over which p(s) is normalised – an effect of the Markov nature of the soil moisture process. properties, impact of parameters on properties etc Note: Equilibrium distribution is for linear evapotranspiration

18. Impact of climate, soil and vegetation on equilibrium distribution Parameters chosen to represent a) tropical climate and vegetation, frequent moderate rainfall, deep soil; b) hot arid region, shallow sandy soil, mixture of trees and grasses; c) cold arid region; d) forested temperate region.

19. Spatial-temporal soil moisture Soil moisture is spatially dependent, because of • correlated rainfall input • ground topology causing run-off from one location to affect nearby locations • correlated vegetation cover We assume • a stochastic process of rain cells with random spatial extents • a flat landscape to avoid run-off problems, eg savannah • a) a homogenous vegetation, or b) a stochastic process of trees with random canopies in a grassy landscape

20. The simplest model… • temporally instantaneous rainfall (ie daily timescale) at random times Tk • linear losses (hot arid region, cf Fig (b)) • ( will be vegetation and soil-dependent) • ignore bound on soil moisture • In this case • proportional interception • standardised infiltration for rainfall • heterogeneoussoil and vegetation • and depend on location

21. Equilibrium distribution for…. hot arid region, shallow sandy soil, mixture of trees and grasses; s* = 0.45

22. Shot-noise process S(t) t Linear losses ( ) and no saturation exponential decay: if there is no input in (0,t). In this case and S(t) has no atom at 0.

23. Rainfall process… • Poisson process of rain cell origins, rate in space-time • circular cells, random radii (iid) • rainfall is instantaneous in time over the cell, depthsY(iid) • at a fixed location, A say, rain events occur in a temporal Poisson process of rate • events occur at locations A and B, d apart, in a temporal Poisson process of rate • Here • is the area of overlap of two unit discs, centresu apart.

24. Marginal distribution … • Transient distribution and its properties • Equilibrium distribution • where is the mgf of the rain depth Y, with • If Y ~ exp( ), S ~ • For general infiltration, replace integrand by • where is the mgf of infiltration from a rain depth Y

25. Joint distribution: sites A and B, d apart… • SA(t) - rain events before t that only affect A, rate • - rain events beforet that affect both A and B, rate • SB(t+h) - events before t+h that only affect B, rate • - events in (t, t+h) that affect both A and B, rate • - events before tthat affect both A and B, rate • Properties of transient distribution • Equilibrium distribution

26. Joint equilibrium mgf: In particular For general infiltration

27. Alternative model: rain cells with exponential durations… • As before, assume • Poisson process of rain cell origins, rate in space-time, and circular cells, random radii (iid) • Assume • rain cell durationD, with constant intensityV(iid) • Observe their superposition where • Soil moisture • (assuming, as before, linear losses, proportional interception and ignoring bound on soil moisture)

28. formal solution… In particular, the covariance properties of (assuming D~ exp( ) ) imply those of S (via Campbell’s Th) The corresponding covariance for the pulse rainfall model is

29. Properties for homogeneous vegetation • Correlation as a function of the spatial and temporal lags • Effect of spatial averaging (different spatial scales). Analytic results can be obtained by using a Gaussian approximation to • Effect of spatial and temporal averaging (different scales)

30. Correlation as a function of spatial and temporal lags (rainfall parameters fitted to data from 17 gauges in Southern Italy, two values for soil porosity-root depth factor) nZr=100mm nZr=500mm

31. Standard deviation of spatially averaged field relative to standard deviation at a point Here is the mean rain cell radius.The ratio depends only on and the spatial area

32. Standard deviation of spatially and temporally averaged fields

33. Heterogeneous vegetation – trees in a grassy landscape • A model for tree crowns….. • Poisson process of tree locations, rate in space • Circular canopies, random radii (iid) • No. of trees covering location, A say, • No. of trees covering A and B, d apart, • P(neither A nor B covered) • P(A is covered, B is not) • P(both A and B are covered)

34. a realisation of the vegetation process…

35. Use probabilities to remove conditioning of previous results on vegetation cover, and determine corresponding properties with random vegetation eg variance of spatially integrated soil moisture

36. Biomass and soil moisture…temporal process For water-limited ecosystems, a simple model for the coupled system of biomass B and soil moisture Sis Assume (within a growing season)

37. transient solution… moments, eg Equilibrium: (deterministic)

38. Summary and scope for further work • Single-site, temporal models of soil moisture • Spatial-temporal models of soil moisture • *simplifications - flat landscape • - linear evapotranspiration • - ignore bound at s= 1 • *spatially-distributed rainfall • instantaneous distributional results • temporally distributed second order results (proportional interception only) • variable vegetation • areally-averaged properties • Coupled dynamics of biomass and soil moisture • *single site, temporal process