Available online at www.sciencedirect.com

Procedia Environmental Sciences 19 (2013) 26 – 36

Four Decades of Progress in Monitoring and Modeling of Processes in the Soil-PlantAtmosphere System: Applications and Challenges

Bayesian inference of tree water relations using a Soil-TreeAtmosphere Continuum model
J. Ringsa, T. Kamaia, M. Kandelousa, P. Hartsougha, J. Simunek
kb, J. A. Vrugtc,
aand J.W. Hopmans *aDepartment of Land, Air and Water Resources, University of California, Davis, CA, USAb
Department of Environmental Sciences, University of California, Riverside, CA, USAc
Department of Civil and Environmental Engineering, University of California, Irvine, CA, USA

Abstract
To better understand root-soil water interactions, a mature white fir (Abies concolor) and the surrounding root zone
were continuously monitored (sap flow, canopy stem water potential, soil moisture, and temperature), to characterize
tree hydrodynamics. We present a hydrodynamic flow model, simulating unsaturated flow in the soil and tree withstress functions controlling spatially distributed root water uptake and canopy transpiration. Using the van Genuchten
functions, we parameterize the effective retention and unsaturated hydraulic conductivity functions of the tree
sapwood and soil, soil and canopy stress functions, and radial root zone distribution. To parameterize the in-situ treewater relationships, we combine a numerical model with observational data in an optimization framework,
minimizing residuals between simulated and measured observational data of soil and tree canopy. Using the MCMC
method, the HYDRUS model is run in an iterative process that adjusts parameters until residuals are minimized.
Using these optimized parameters, the HYDRUS model simulates diurnal tree water potential and sap flow as afunction of tree height, in addition to spatially distributed changes in soil water storage and soil water potential.
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Keywords: hydraulic properities; tree ET; numerical modeling; parameter optimization

1. Introduction
Trees and forests play a key role in controlling the water and energy balance at the land-air surface. By
way of biophysical principles, soil water is taken up by the roots and moves through the water-conducting* Corresponding author. Tel.: 530-752-3060.
E-mail address: jwhopmans@ucdavis.edu.

1878-0296 © 2013 The Authors. Published by Elsevier B.V
Selection and/or peer-review under responsibility of the Scientific Committee of the Conference
doi:10.1016/j.proenv.2013.06.004

J. Rings et al. / Procedia Environmental Sciences 19 (2013) 26 – 36

vessels or xylem along a water potential gradient into the canopy, where it transpires into the atmospherethrough leaf stomata. The changing soil water content affects both hydrological and tree responses, by
way of regulating stream flow transpiration rates, and ecosystem functions.
Numerical water flow models allow simulating the relevant soil hydrological processes in their mostbasic form, such as water infiltration, soil water redistribution and deep percolation, as well as water
uptake by roots and the effects of soil water stress on tree transpiration. Early attempts to model treewater flow were often based on the electrical circuit analog [1], with resistance-capacitance (RC) models
to characterize xyleme resistance to water flow and water storage, and allowing for a time lag between
sap flux density and transpiration rate [2,3]. Subsequent models of varying complexity were introduced
using the RC approach [4,5,6], however, problems arise to fully describe the measured spatio-temporalvariations in tree hydraulic conductance and water storage, as controlled by tree water potential, with thechanging xylem saturation caused by partial cavitation of the tree conducting vessels [7].
Though models were improved, for example by considering hydraulic capacitance being a function ofwater potential, more recent studies have abandoned the RC approach and instead embraced thephysically-based nonlinear modeling framework [7,8,9,10,11,12], as typically applied for unsaturated
water flow modeling [13,14]. These numerical flow models discretize the modeled domain into small
elements and use finite-difference or finite-element methods to solve the flow equations. However, few
models use the integrated soil-tree continuum approach, coupling the soil with the tree domain,
simulating the soil, roots and tree trunk as a continuum, in which water flow is driven by water potential
gradients along the coupled Soil-Tree-Atmosphere-Continuum or STAC [12,15] with spatiallydistributed root water uptake and canopy transpiration sink terms.
By way of the coupled hydrodynamics flow model at the tree scale, water flow through the tree rootsystem and stem is driven by the evaporative demand and soil-available water, leading to a gradient inxylem water potential along the STAC. Water flow through the coupled system is described using theequation [16], with both the soil and tree conducting domains modeled asa porous medium, defined by nonlinear soil and tree water relationships. As part of the STAC, thecoupled model computes the spatial-temporal changes inthe xylem water potential, xylem water content
and xylem water flux or sapflow, along the tree trunk. Potential tree transpiration is reduced by a water
potential dependent reduction function to determine actual tree transpiration. By defining a tree trunkwater capacity term, the model is able to simulate the dynamic behavior of tree hydraulics, including thecontrol of tree water storage and xylem conductivity on tree transpiration.
Bittner et al. [15] were among the first to apply the coupled soil-tree porous media model to simulate
tree hydrodynamics by solving the mass-conserving PDE or Richards model, that solves for the spatialtemporal dynamics of the tree hydraulic system, from which tree transpiration can be determined fromsimulated sap flux densities as computed from xylem axial conductances and water potential gradientsalong the tree stem and canopy, and including a sink term to represent spatially-distributed canopy
transpiration. Tyree [5] already showed the importance of including tree hydraulic architecture andgeometry with corresponding hydraulic conductance and stem water capacitance on matching measured
tree water dynamics, controlling tree response to changing soil water conditions, with tree trunk water
storage and changing unsaturated conductivity caused by cavitation of the water-conducting vessels in thesap wood [7]. Bitter et al. [17] demonstrated that differences in stomatal closure and drought tolerance
between tree species is caused by differentiation in tree xylem properties between species.
Soil tree hydraulic parameters can be measured in the laboratory, however these measurements are
difficult and mostly destructive. For the purpose to achieve representative tree water relationships theirparameters must be calibrated by comparison of simulated with observational data. Alternatively, as has
been done for soils [18,19], effective hydraulic parameters can be obtained through an optimization
framework, whereby residuals between measured and model-simulated variables are minimized. The

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J. Rings et al. / Procedia Environmental Sciences 19 (2013) 26 – 36

main objective of our paper is to estimate the tree hydraulic relationships and their control on tree water
stress and transpiration, using a global parameter optimization technique.
2. Site description and measurements
Measurements were conducted in and around a White Fir (Abies concolor) in a 99 ha subcatchment(P301) of the King's River Experimental Watershed (KREW), located in the rain-snow transition zone of
the southern Sierra Nevada mountain range in California at an elevation of 2018m. This study is part of
the Southern Sierra Critical Zone Observatory (CZO), established to improve understanding of surface
and subsurface processes along a gradient of elevation, energy, water, and soil [20,21]. The White Fir(CZT-1) is located along a ridge in a relatively open area of the forest at an elevation of 2018 m (Bales etal., 2011b), near a meteorological station. The mature tree is about 30 m high, with a trunk diameter of
0.55 m.
Undisturbed soil samples were collected to a depth of 2.5m for soil analysis. The top 1 m soil was aloamy sand to sandy loam with a range of sand content from 70-84 %, with sand content increasing from84-90 % below the 1 m soil depth. Dry bulk density values ranged from about 1.0 g cm-3 (15 cm
sampling depth) to about 1.4 g cm-3 (60 and 90 cm sampling depth) and increasing to near 1.5 g cm-3 at
the 2.5 m depth. Corresponding measurements of saturated hydraulic conductivity using the constanthead method [22] showed values between 5-10 cm/hr in the top 1 m, decreasing to around 1 cm/hr in the
1-2.5 m depth interval. Soil hydraulic parameters for soil samples of the top 1 m were obtained using themodified multistep outflow method as presented by Nasta et al. [23]. Excavations further down below 2.5
m indicated the presence of weathered bedrock that changes with depth to moderately dense saprolite andconsolidated saprock and hard bedrock below 5m.
Data used in the presented study include soil moisture, tree sap flux and stem water potential
measurements, with the meterorological data of the flux tower providing the necessary information forpotential tree evapotranspiration. Data were selected for an 18-day rainless period in summer of 2009,
starting July 15. Echo- 5TE soil moisture sensors were installed at depths of 0.15 m, 0.30 m, 0.60 m and0.90 m in each of 6 locations within a 5 m radius from the tree trunk. The sensors were calibrated in thelaboratory [24], from which it was determined that their accuracy is around 3% for a range of soils. Using
all water content measurements, average total soil water storage (m3) was computed during the 18-day
measurement period every half hour. Three sapflow sensors (TransfloNZ, Palmerston North, NZ) were
installed into the sapwood at a trunk height of 2.5m. Using the compensation heat pulse technique (Greenand Clothier 1988), average sap flux flow (L/d) was estimated at 30-minute time intervals. All soil
moisture and sapflow data were stored by a data logger. Stem water potential measurements were taken[25] from needle stems of various lower tree branches, at about 6 m from the ground. Sevenmeasurements were taken during 24 hours on July 21-22, 2009.
3. Soil-Tree-Atmosphere-Continuum Model
By way of numerical flow modeling, the overall goal is to identify the physical processes andhydrological properties that control spatial and temporal changes in water flow of the coupled soil-treeatmosphere domain. The requirements for this model are to (a) simulate time series of tree sap flow andstem water potential, soil water content and matric potential, and (b) to couple collected data withnumerical flow simulations, allowing estimation of tree water relationships (water retention andunsaturated hydraulic conductivity functions). The STAC model is driven by spatially distributed
potential evaporation in the canopy and includes reduction functions of potential evapotranspiration (ET)and root water uptake as determined by leaf and soil water potential, respectively. In order to estimate

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J. Rings et al. / Procedia Environmental Sciences 19 (2013) 26 – 36

deep soil water contributions to tree ET, the soil domain was extended to 5 m depth. The following model
was developed for the main purpose to characterize and identify the main physical processes that control
tree response to limiting soil water availability conditions, requiring coupling of soil with tree domains
using the unifying concept of water potential that drives water through the STAC.
3.1. Unifying Equations
Across the soil-tree domain, unsaturated water flow is described by the Richards equation [16], as
presented here in its axisymmetrical, two-dimensional, isotropic form
(1)
where is the volumetric soil water content (m3 m-3), k defines the unsaturated hydraulic conductivity
function (md-1), h is the soil water pressure head (m), r is the lateral coordinate, z is the verticalcoordinate (positive downwards), t is time (T) and W(h,r,z) defines a sink/source term (m3 m-3 d-1) thatquantifies either spatially distributed root water uptake (soil, WSoil) or canopy transpiration (tree, WTree).
Both k and W are functions of and/or h. The subscripts r and z allows for the possibility to include soil
anisotropy, i.e., to simulate water flow with the unsaturated hydraulic conductivity function being
different for the r- and z-direction.
For solution of Eq. [1], the water retention and unsaturated hydraulic conductivity functions must be
defined for both the soil and tree conducting matrix, with the retention function characterizing water
storage capacity. We define these functions using the relationships of [27,28], using the degree ofeffective saturation, Seff, or
(2)
and
(3)
where s denotes the saturated water content at h = 0 (m3 m-3), r is the residual water content (m3 m-3),= s (m d-1).
and n (m=1-1/n) are shape parameters, and ks is defined as the saturated conductivity at
3.2. Domain Definition, Boundary and Initial Conditions
We use the finite element HYDRUS software [14], to solve the Richards Eq. (1) in a discretizedsystem of linear equations, for water potential, volumetric water content and water flux density across thecoupled soil-tree domain and assuming axial symmetry around the tree trunk. Both the soil and tree trunk
are modelled as axial-symmetrical, represented by a rectangular domain (Fig. 1). The simulated soil
domain extends to 5 m outwards, and is 5 m deep with three soil layers characterizing the unsaturated
soil, with the 2.5-5 m depth interval representing the weathered low conductivity saprolite that can store
water but is inaccessible to tree roots. The 10 cm radius of the tree trunk domain was chosen so that thedomain volume is approximately equal to that of the sapwood of the real tree. Model solution also
requires knowledge of the initial water potential distribution across the soil-tree domain. For the soil
domain we converted the 24 soil moisture data of July 15 to soil water matric potential values using thelaboratory-measured retention curves, and applied a 2nd order polynomial interpolation scheme toestimate the soil water potential across the measured soil domain, assuming hydraulic equilibrium at thedomain boundaries.

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J. Rings et al. / Procedia Environmental Sciences 19 (2013) 26 – 36

Fig. 1. Schematic of simulated SPAC domain.

Meteorological data from the weather tower were used to estimate local hourly potential
evapotranspiration, ETo , using the Penman-Monteith equation [29]. Values for the aerodynamic
resistance and bulk surface stomatal resistance terms were calculated according to FAO guidelines (Allenet al., 1998). To estimate the potential tree transpiration, ET p, we adopted the crop coefficient approach[30], and multiplied ETo with a tree coefficient, sETo (-) , or
ETp = sEToETo(4)

Moreover, because of the soil dryness for the selected monitoring period, we could assume that soil
evaporation was nil. The lower boundary of the soil at the 5 m depth was described by a seepage
boundary, allowing water to leave the soil domain when saturated, and more importantly allowing forboth upwards and downwards flow across the whole soil domain.
3.3. Coupling of root water uptake with tree canopy transpiration.
Root water uptake is simulated by the W-term in Eq. [1], representing actual uptake (m3 m-3 day-1) ateach node in the soil domain, controlled by root density distribution, Soil (r,z) (m-3), and a soil water
stress response function, Soil (h). Both have functional values between 0 and 1. For an axisymmetricalsoil domain , the normalized root water uptake distribution, Soil (r,z) is defined by [31,32]:(5a)
as defined by [31]:with a general nonuniform root distribution,* r, z

1

z
z

m

1

r
r

e

pz *
z z
zm

pr *
r r
rm

(5b)

m

*

where rm and zm define the maximum rooting extent in the radial and depth direction (m). z and r* areempirical parameters (m) that shift the maximum of the distribution in vertical and radial direction,

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J. Rings et al. / Procedia Environmental Sciences 19 (2013) 26 – 36

respectively, and pz and pr are empirical parameters that determine the exponential shape of the
distribution.
For water-stressed soil conditions, the actual root water uptake term in Eq. (1) is computed from:(6)
where soil(h) (dimensionless) was introduced by Feddes et al. [33], and reduces root water uptake from its
maximum possible value because of soil water stress. The Feddes function is defined by four water
potential values, P1 through P4. For soil water potential values between P 2 and P3, soil(h) will be optimum
and equal to 1.0. For h-values between P1 and P2 (soil aeration stress) and between P 3 and P4 and lower(soil water stress), soil values will be smaller than one and zero, at a minimum ( h < P4). Finally, from
integration of Eq. (5) over the soil domain, the actual total root water uptake, Ra, (m d-1) is computed
from:
(7)
In the presented coupled domain, the volume of water taken up by the roots must now be transported in
the conducting vessels (xylem) of the sapwood in the tree trunk. For that purpose, the coupled model
includes a small storage reservoir to simulate the root domain that temporally holds it, and empties it into
domain by way of defining a lower flux boundary condition for the tree domain..
the treeTo simulate tree transpiration, we introduce a sink term, W(h,z), which is a function of the onedimensional canopy density distribution function, tree (z), and a canopy water stress response function,-1tree (h). We define the canopy distribution (m ) by:,,(8)

for z > 6 m and zero otherwise. After normalization, and defining a canopy stress function that is of asimilar Feddes form as used for characterizing soil water stress, the actual tree transpiration (ETa) is
computed from:(9)
We realize that Ra = ETa , only if the tree trunk water storage is not changing over the time period of thecalculation. In all, this approach clearly shows the coupling of root water uptake with tree transpiration,
by way of the stress response functions for root water uptake and canopy transpiration, which are definedby soil water potential and stem water potential, respectively.
4. Parameter optimization
The presented continuum model includes numerous parameters, of which we selected 18 of them to be
unknown, apriori, to be estimated using the MT-DREAM optimization method [34]. Soil water retention
and unsaturated hydraulic conductivity parameters of Eqs. [2] and [3] for the top two soil layers wereks and s values foreach layers were considered unknown parameters. We assumed the soil hydraulic parameter values forthe third 2.5-5 m layer to be equal to those of the second layer, except the ks value was reduced by twoorders of magnitude. All tree hydraulic parameters ( tree, ntree , ks,tree and s,tree) were assumed unknown
and were part of the optimized parameter set. The residual water content value ( r,tree ) was set to zero.

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J. Rings et al. / Procedia Environmental Sciences 19 (2013) 26 – 36

Maximum rooting extend in the vertical (zm = 1.6 m) and lateral direction (rm =5.0 m) were known
from root excavation data for a similar nearby tree, but all other four parameters (z*, r*, pz , and pr) were
optimized. Stress response function parameters, P1 through P4, were partially assumed known. For thesoil water stress function we only optimized P3, whereas other values were fixed to P1,soil = -1 kPa , P2,soil= -2.5 kPa, and P4,soil = -2,000 kPa. For the canopy water stress response function, both values of P3 andP4 were optimized, with fixed values of P1,tree = -1 kPa , P2,tree = -0.2 kPa. Finally, we introduced a sap
flow scaling parameter, ssap, to adjust the magnitude of measured sap flow rates to those required frommeasured soil water storage changes, or = ssap qsap, measured,(10)

as sap flow measurements are known to accurately represent tree sap flow dynamics, however, theirmagnitudes are typically uncertain.
We applied a global Markov Chain Monte Carlo (MCMC) optimization framework that allows for (a)large number of unknown parameters, (b) different measurement types, and (c) quantification ofparameter uncertainty. Specifically, we used the DiffeRential Evolution Adaptive Metropolis (DREAM)
family of algorithms [34,35,36,37]. To find the best possible agreement between a series of N
observations Y = Y1 . . . YN and N simulated values y = y1 . . . yN, the parameter optimization techniqueminimizes the objective function, SS, that contains the sum of squares of residuals between simulated
and measured variables, or(11)This is accomplished through iterative solution of the hydrodynamic STAC model, changing parameters
MCMC methods guide Markov chains on a random walk throughof the optimized parameter vector,
the parameters space. The final optimization result includes a posterior distribution of optimized values
from which parameter uncertainty can be derived. The MTfor each of the optimization vector,
DREAM algorithm optimizes the parameter in parallel through multiple chains of parameter sets, and it istherefore that the solution is most efficiently run on a computer cluster.
5. Results and Discussion
An important result of parameter optimization is the correlation between optimized parameters. Thatis, high parameter correlation is an indication of nonuniqueness between the correlated parameters, andlikely will also present itself by a wide postiori and multimodel distribution of those parameters. We
found strong correlation between the parameters tree and ntree , as also determined by Huisman et al., [38]and between ks,tree and the sapflow scaling factor, ssap. The final optimized tree water relationships arepresented in Fig. 2a (tree water retention curve) and Fig. 2b( tree relative conductivity curve). The
optimized curves using the median of the optimized parameter distributions are represented by the green
solid lines, with the blue area characterizing the uncertainty as defined by the 95% confidence interval of
the posterior distribution, as calculated by a bootstrapping technique.
The tree water retention curve agrees well with literature data [39, 40] The agreement with literature
values for the relative hydraulic conductivity curve is not as good, but is especially very good whencompared with the Waring and Running [39] experimental data for a white fir (orange crosses).However,in all honesty, we should not always expect agreement, as referenced experimental tree hydraulic data are
typically collected from small tree branches or wood cuttings, whereas our tree water relationships
represent effective hydraulic functions for the sapwood of the whole tree.

J. Rings et al. / Procedia Environmental Sciences 19 (2013) 26 – 36

Fig. 2. Comparison of independently measured (symbols) with optimized tree water relationships, (a) tree water retention curve; (b)tree hydraulic conductivity functions

Comparison of measured and simulated hydrologic variables are presented in Fig. 3, for ETp (cm/d),
sap flux (L/h), stem water potential ( kPa) and soil water storage (m3) for the simulated 18-day period ofJuly, 2009. The ET data in the top panel compares potential ET using the flux tower weather data (blue)
with the scaled ET (ETp) using the tree coefficient, sETo. Given the uncertainty of the data, the model fit isan excellent agreement with soil and tree observations.

Fig. 3. Comparison of measured with simulated ETp, sap flow, stem water potential, and soil water storage

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J. Rings et al. / Procedia Environmental Sciences 19 (2013) 26 – 36

Figure 4 presents diurnal fluctuations in tree water potential (left panel) and sapflow (right panel) for aperiod of 48 hours during the 18-day simulation period, at 10 heights, from the base of the tree (1 m) to
the top of the tree at 28 m.
As expected, diurnal fluctuations increase as one moves from the base to the top of the tree, as
controlled by the reduced water delivery by sapflow upwards into the tree canopy. Canopy stress ismostly felt towards the top of the tree canopy, with stem water potential approaching -2,000 kPa. We notethat the optimized P3,tree value was -1,500 kPa, indicating that the tree was under water stress during partof the day.
We also note the recovery of stem water potential overnight, as determined by the refilling of thewater-conducting vessels (right panel, Fig. 4). As is intuitively clear, sapflow decreases from the tree basetowards the top of the tree because of tree transpiration. The largest diurnal fluctuations in sap flow occur
near the base of the tree with high sap flow values during the day, decreasing towards zero overnight, andrefilling the water conducting sap wood in the tree trunk. The magnitude of the diurnal fluctuations iscontrolled by both atmospheric forcing and the tree water relationships. The tree water retention curve
controls the water capacity of the tree and the temporal variations in tree water potential and sap watercontent, whereas the relative conductivity functions controls sap flow rate and water delivery to thetranspiring canopy, as well as the rate of water redistribution, replenishing the decreasing water storage inthe water-stressed sapwood and canopy.

Fig. 4 Simulated sap flow (left panel) and corresponding stem water potential (right panel) for 10 tree heights for a 48-period

Acknowledgements
Research was supported by the National Science Foundation, through the Southern Sierra Critical
Zone Observatory (EAR-0725097) and a Major Research Instrumentation grant (EAR-0619947).
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