Statistical Optimization of Hydraulic Parameters of a

Soil-Tree-Atmosphere Continuum Model of a Sierra

Nevada White Fir

J. Ringsa, T. Kamaia, M. Kandelousa, P. Nastab, P. Hartsougha, J.Simunekc, J. A. Vrugtd, J. W. Hopmansa,∗

aDepartment of Land, Air and Water Resources, University of California, Davis, CA,

USAbDepartment of Agricultural Engineering and Agronomy, University of Napoli Federico

II, Naples, ItalycDepartment of Environmental Sciences, University of California, Riverside, CA, USAdDepartment of Civil and Environmental Engineering, University of California, Irvina,

CA, USA

Abstract

TEXT

Keywords: Fire, Ice, Air

1. Introduction1

Trees play a key role in controlling the water and energy balance at the2

land-air surface. As a physical medium, they transport water from the soil3

along a water potential gradient to the canopy, where it is transpired into4

the atmosphere. Thus changing water content of soil and atmosphere, they5

influence meteorological, climatological and hydrological cycles.6

Numerical models allow simulating the relevant processes, most importantly7

∗J. W. Hopmans( )

Email address: jwhopmans@ucdavis.edu (J. W. Hopmans)

Preprint submitted to Ecological Modelling March 5, 2012

the transport of water as it enters the soil, is transported through the soil,8

taken up by roots into the tree and ultimately transpired into the atmosphere.9

Early attempts to model tree water flow were often based on an analogy to10

an electrical circuit proposed first by Van den Honert (1948). The main11

motivation for drawing on this analogy is a time lag between sap flux and12

transpiration rate that is caused by the tree’s capacity to store water (Waring13

and Running, 1978; Jarvis et al., 1981), and the resistance of the xylem to14

water flow. Models of varying complexity have been introduced (e.g. Cowan,15

1965; Tyree, 1988; Da Silva et al., 2011), but a number of problems have16

been identified. This approach has limitations modeling water flow since it is17

inadequate in separating changes in saturation change from changes in pres-18

sure (Aumann and Ford, 2002), and thus fails e.g. to consider the influence19

of saturation on hydraulic conductivity (Kramer and Boyer, 1995). Another20

problem is that the electrical circuit model allows for negative values, since21

electrical charges can be positive and negative. Water content, however, is22

strictly positive, nut nothing in the model inheritly prevents it from becom-23

ing negative (Chuang et al., 2006). Finally, physical flow processes in sap24

wood are are a multi-phase flow problem, since both air and water interact25

in the xylem vessels (Aumann and Ford, 2002).26

Attempts have been undertaken to alleviate these problems, e.g. Frueh and27

Kurth (1999) draw from a connection to hydrological models and treats trees28

as a porous medium (Siau, 1984) by considering hydraulic capacitance vary-29

ing with pressure. However, more recent studies have fully given up the30

electric circuit analogy as advocated by Fiscus and Kaufmann (1990) and31

embraced physics based hydrological models (Kumagai, 2001; Aumann and32

2

Ford, 2002; Bohrer et al., 2005; Chuang et al., 2006; Siqueira et al., 2008;33

Janott et al., 2011; Kamai et al., 2012).34

These models discretize the modeled domain into small elements and use35

finite-difference or finite-element methods to solve the flow equations. Of36

the models presented, some concentrate on only the canopy (Bohrer et al.,37

2005; Chuang et al., 2006) or only the subsurface (Siqueira et al., 2008). A38

model considering soil, roots and tree as a continuum has been presented by39

Janott et al. (2011), but the tree is modeled as a simple chain of cylindrical40

elements. In this study, we make use of a recently developed model (Kamai41

et al., 2012) that describes the full continuum soil-root-tree in a finite-element42

model, with simulated unsaturated flow (Richards, 1931) in the soil and tree43

and stress functions (Feddes et al., 1978; Vrugt et al., 2001b) determining44

spatially distributed root uptake and canopy transpiration sink terms.45

To fully describe the flow processes, we must parameterize water reten-46

tion and unsaturated hydraulic conductivity functions (Mualem, 1976; van47

Genuchten, 1980). To make a model representative of nature, the parameters48

of these functions have to be calibrated by comparison to observational data.49

While parameters can also be measured e.g. in the laboratory, these measure-50

ments are mostly destructive (e.g. Heine, 1970; Zimmermann, 1978; Waring51

and Running, 1978; Cochard, 1992a; Sperry and Ikeda, 1997; Li et al., 2008).52

Better understanding of the system soil-root-tree-atmosphere, however, can53

be gained from monitoring of hydrologic processes in soil and tree.54

To this end, non-destructive measurement techniques exist to ressolve the55

temporally and spatially variable hydrologic states of the soil-tree-atmosphere56

system. A range of techniques allow measuring the water content of the soil,57

3

with point probes methods like time domain reflectometry (TDR) (Noborio,58

2001; Robinson et al., 2008), capacitive probes (Bogena et al., 2007; Kizito59

et al., 2008) and heat pulse probes (Mori et al., 2003; Kamai et al., 2009). In60

recent years, hydrogeophysical methods have increasingly been established61

that use geophysical techniques to retrieve water content distributions with-62

out disturbing the soil (see e.g. Huisman et al., 2003; Rubin and Hubbard,63

2005; Vereecken et al., 2006; Rings et al., 2008). The water uptake of the64

tree can be monitored by sap flux sensors (..), while the water potential of65

the tree is retrieved from leaf water potential measurements (...).66

67

The goal of this study was to implement an optimization framework that68

allows the identification of the hydraulic parameters of the tree, along with69

parameters of the soil and root models. Optimization can be understood as70

an iterative process that adjusts parameters until the smallest discrepancy71

is found between modeled and measured observational data. Data was col-72

lected on a White Fir in a mountaineous environment in the Sierra Nevada.73

There are several reasons for using a modern statistical optimization frame-74

work: (a) the large number of parameters enforces the use of a powerful75

method that is able to handle complexly shaped objective spaces, (b) sta-76

tistical inference should be seen as a tool to learn about the model and the77

natural processes, e.g. by appraising model and measurement errors as un-78

certainty in the optimized parameters, and (c) modern computer clusters79

and state-of-the-art MCMC algorithms (Vrugt et al., 2008a, 2009a; Laloy80

and Vrugt, 2011) provide the technical opportunity to implement such a81

framework.82

4

2. Site Description83

The tree that is modeled in this study is a White Fir in the King’s River84

Experimental Watershed (KREW) in the Southern Sierra Critical Zone Ob-85

servatory. Measurement operations in the observatory target a better under-86

standing of water cycles in a changing environment, focussing on the western87

US mountains with its forest landscapes that are controlled by either rain or88

snow patterns. The measurements also focus on the interaction and feedbacks89

between landscape, hydrology and vegetation. In this context, Bales et al.90

(2011) describe a prototype instrument cluster that studies water balance91

on the boundary between rain and snow-dominated areas. The instrument92

portfolio contains central instruments like a meteorological station with eddy-93

covariance (flux tower) and soil water content, snow depth, matric potential94

and evapotranspiration monitor probes. In addition, stratified sampling cap-95

tures landscape variability.96

In the vicinity of the flux tower and meterological station, a White Fir (Abies97

concolor) tree was fully instrumented to improve understanding of the root-98

water-tree system in the context of the watershed experiments. The site is99

characterized by a shallow soil of about 2.5 m depth, underlain by saprolite100

and granite bedrock (...). Since the summer of 2008, more than 250 sensors101

were deployed to capture soil moisture content, matric potential and tem-102

perature, sap flux, stem water potential in the canopy to capture seasonal103

cycles of soil wetting and drying.104

Figure 1 shows schematics of the site set-up with blue circles marking the105

six pits containing four Decagon EC5 soil moisture sensors each at depths106

of 15 cm, 30 cm, 60 cm and 90 cm used in this study. The tree itself is107

5

instrumented with a heat pulse sap flow sensor (...). There is uncertainty108

in the amplitude of the sap flux, as the values for 2009 and 2010 differ by a109

factor of 3. To circumvent this problems, we introduce a sap flux amplitude110

scaling factor ssap. We estimate this factor in the optimization and divide all111

modeled sap flux values by ssap. This way, we still make use of the time series112

dynamics, which should contain a vital part of the information required to113

constrain the hydraulic parameters of the tree (??maybe move somewhere114

else??).115

On some dates, leaf water potential was measured by ...116

We use data collected in 2009 and 2010 for two optimization scenarios. The117

first set encompasses 18 days starting July 15th 2009, without recorded rain-118

fall at the end of a dry season. Sap flux was recorded every 30 minutes, soil119

moisture hourly and on July 21st and 22nd 2009, stem water potential was120

measured seven times over the course of 24 hours.121

For the second data set, we use a period of 20 days starting October 2nd,122

since there were two major rainfall events during this period. Unfortunately,123

only sap flux and soil moisture measurements are available for this second124

data set.125

3. Soil-Tree-Atmosphere Continuum Model126

The flow of water is modeled as a continuum through the subdomains127

soil, roots and tree. We represent the model domain in a two-dimensional128

model that is assumed to be axisymmetrical around the center of the tree.129

Although the measurement pits in the soil are placed in different circle seg-130

ments around the tree, this assumption allows us to include the six pits in131

6

the two-dimensional model at their radial distance from the tree.132

Atmospheric conditions force water movement, either by evapotranspiration133

from the soil and tree canopy into the atmosphere, or by providing rainfall134

that infiltrates into the soil. Figure 2 shows the setup of the model domain.135

The soil is divided into three layers, with the deepest layer having a low136

hydraulic conductivity. It models the saprolite layer between the upper soil137

and rock, which can store water but is not accessed by roots. The two upper138

soil layers have different material properties. The water taken up by roots139

Ra is routed through a flux boundary into the second subdomain, the root140

buffer. The root buffer is modeled as an infinitely conductive material, so141

that the tree can take up the water as it needs it and the water balance is142

kept (Kamai et al., 2012). The third subdomain, the tree, is modeled as a143

rectangular domain for computational efficiency. Its size has been chosen so144

that the volume of the axisymmetric model tree domain is equal to that of145

the sapwood of the actual, conical tree.146

Figure 3 gives an overview of the different water fluxes in the model. Tower147

data (see section 3.2) serves as the basis for obtaining the tower evapotran-148

spiration ET0 by applying the Penman-Monteith equation. This could be149

used as a basis for estimating bulk stomatal conductance (Baldocchi et al.,150

1991), however this would introduce a large number of additional unknown151

parameters that might not be well constrained by the available measure-152

ments. Rather, Kamai et al. (2012) have introduced a site-specific factor153

sET0 to obtain potential evapotranspiration ETp from scaling of the tower154

data.155

We set both the potential root uptake Rp and potential transpiration from156

7

the tree Tp equal to ETp. The actual root uptake Ra is routed into the root157

buffer, which the tree accesses. To remove the transpiring water Ta from the158

tree, sink terms are placed along mesh elements in the tree domain. Section159

(3.3) will describe in more the model parts that determine root uptake and160

canopy transpiration.161

3.1. Hydrological Model162

To solve the equations of water flow in the model above we use an adapted163

version of HYDRUS (Simunek et al., 2008) that enables routing the root wa-164

ter uptake into the root buffer, and to model the canopy transpiration with165

a second Feddes model.166

HYDRUS solves the Richards equation (Richards, 1931) for unsaturated wa-167

ter flow using a linear Galerkin approach in a finite element scheme. We168

assume that the parametric model by van Genuchten (1980) can describe169

soil water retention and hydraulic conductivity as170

S =θ − θrθS − θr

= (1 + |αh|n)−m (1)

171

k(h) = kS√S[

1−(

1− S1/m)m]2

(2)

where S is water saturation, h is pressure head (cm), θ is the volumetric172

water content (cm3/cm3), θr is the residual water content for h→−∞, θS173

is the saturated volumetric water content, α is the inverse of the air-entry174

value (cm−1), n is an empirical shape factor (–), m is a factor that can be175

connected to n via m=1−1/n and kS is the hydraulic conductivity at θ=θS176

(cm/d).177

The hydraulic parameters of the soil layers are partially determined from178

8

laboratory measurements, but the measurement are preliminary and uncer-179

tain. Therefore, some estimation of soil hydraulic parameters has to included180

in the optimization. To limit the number of parameters, we restrict this to181

log kS and θS of either soil layer. The other parameters are set to the labora-182

tory values, and the deep saprolite layer keeps the parameters of the second183

soil layer, but with a hydraulic conductivity decreased by two orders of mag-184

nitude. ??Table with all soil hydraulic parameters??185

The initial conditions are specified in terms of pressure head. For the tree,186

we estimate the pressure head at the base of the tree hini, Tr in the optimiza-187

tion, and then initialize the tree in hydraulic equilibrium from the base. For188

the soil, we first calculate the pressure head at each of the 24 soil moisture189

probe locations, and use a linear 2D interpolation scheme to approximate190

the initial distribution in between the measurement pits. The pressure heads191

at the left, right and bottom boundary of the area covered by probes are ex-192

tended to the edges of the modeled domain, and extended with in hydraulic193

equilibrium above the -15 cm probe. Figure 6 shows the initial pressure head194

distribution for the 2009 period.195

3.2. Atmospheric Boundary196

For one of the two periods simulated, rainfall was measured at (...). As197

this was not measured under canopy, effects of rainfall interception and sur-198

face runoff are unknown. Therefore, we scale the rainfall by a parameter srain199

and include that parameter in the optimization.200

9

3.3. Root and Canopy201

To connect soil and tree domain of the model, water uptake by roots and202

transpiration has to be modeled. In the following, the root water uptake203

model is described, which is based on the idea of introducing a sink term at204

each node of the model mesh. The same idea can be used to model canopy205

transpiration, by introducing a sink term on the nodes of the tree domain.206

The distribution models the canopy shape, therefore sink terms start at 6 m207

and above and linearly decrease with height to the top.208

3.3.1. Root Density Model209

We rely on a two-dimensional root distribution model as developed by210

Vrugt et al. (2001a) as an extension of a one-dimensional model introduced211

by Raats (1974). It expresses root density as the product of a dimensionless212

radial distribution function (β(r)) and a depth distribution function (β(z))213

as214

β(r, z) =

(

1− z

zm

)(

1− r

rm

)

exp

[

−(

pzzm

|z∗ − z|+ prrm

|r∗ − r|)]

(3)

where rm and zm denote the maximum rooting radius and depth, r∗ and z∗215

are empirical parameters that shift the maximum of the distribution in radial216

and depth and rp and zp are empirical factors that determine the shape of217

the distribution.218

Two of these parameters were fixed based on knowledge acquired through the219

excavation of the root system of a nearby white fir stump (Eumont et al.,220

2012). The maximum root depth rm was set to 250 cm based on observational221

digs, and the maximum depth in radial direction was found by Eumont et al.222

(2012) to be 160 cm. The other parameters were added to the optimization.223

10

It could be expected that the maximum radial rooting distance coincides with224

the extent of the modeling domain, however this is not clear from observa-225

tions. Therefore, also zm was obtained through optimization. The resulting226

optimized root density distribution for the 2009 period is shown in Fig. 4227

??Show this? Discuss this later??.228

3.3.2. Root Water Extraction229

The model extracts water from the soil using a sink term that represents230

extraction of water from the soil by roots. In the finite element geometry,231

each node is assigned a sink term. The shape of the sink terms determines the232

proportion of potential root water Uptake Rp actually extracted depending233

on the pressure at each node as:234

Ra(h, r, z) = α(h)β(r, z)Rp (4)

The shape of this water stress response function α(h) as introduced by Feddes235

et al. (1978) is shown in Fig. 5 and is characterized by four characteristic236

pressure values P1 . . . P4.237

4. Optimization Framework238

As seen in the previous sections, we have to deal with (a) a large number239

of unkown parameters, (b) three different measurement types that all should240

be included in the optimization process, and (c) a desire to not only find one241

discrete set of parameters, but to learn about the uncertainty in the param-242

eters.243

To elaborate on the last point, there are various sources of uncertainty in244

11

modeling a natural system, notably structural errors caused by simplifica-245

tions in the way the model approaches the true processes, and errors in246

measuring the forcing and observations. Structural errors typically are con-247

siderably larger than modeling errors (Vrugt et al., 2009b), yet they are very248

hard to quantify, and the task of separating different sources of errors is a249

daunting one (see e.g. Keating et al., 2010, and references therein). However,250

it should be clear that an optimization framework that recognizes uncertainty251

at least implicitly, through precipitation into parameter uncertainty, provides252

more information than a deterministic framework that seeks only an optimal253

parameter set. The question whether a formal or informal framework should254

be used has been contested (Vrugt et al., 2009b; Beven, 2008; Vrugt et al.,255

2008b), but (a) a formal statistical optimization framework that is used in256

a way that merges all uncertainty into parameter uncertainty is used in a257

rather informal way and (b) it still offers the possibility of extending or us-258

ing it to disentangling sources of error.259

Furthermore, as we are dealing with a larger number of parameters and com-260

plex, non-linear relations between them, the objective function we seek to261

minimize will have a very complex shape. If we tried to use classical min-262

imization techniques like gradient descent, there is a remarkable chance to263

get stuck in a local minimum. Did we try different starting positions, we264

would be overwhelmed by the dimensionality of the problem; if we applied265

regularization we would artifically cut out parameters with lower sensitivity266

instead of learning about their uncertainty.267

Consequently, we are looking for a global optimization framework that recog-268

nizes model and measurement error implicitly or, if available, even explicitly.269

12

Markov Chain Monte Carlo (MCMC) methods provide such a framework,270

and here we rely on the current state-of-the-art member of the DiffeRential271

Evolution Adaptive Metropolis (DREAM) family of algorithms (Vrugt et al.,272

2008a, 2009a, 2011; Laloy and Vrugt, 2011).273

4.1. (MT)DREAMZS274

We try to understand the workings of MCMC methods from the objective275

of each optimization: To find the best possible agreement between a series of276

N observations Y = Y1 . . . YN and N modeled values y = y1 . . . yN (the added277

complication of different observation types will be discussed later). Most278

often, the discrepancy between modeled and measured values is described by279

the sum of squares or some variant thereof:280

SS =N∑

n=1

(yi − Yi)2 (5)

The y are produced by a model that calculates the time development of281

the observed states dependent upon a set of parameters θ. By iteratively282

changing the parameters, we modify the modeled values and can minimize283

Eq. 5 if we discover good set of parameters.284

MCMC methods guide Markov chains on a random walk through the param-285

eters space. Candidate positions for a chain to jump from position θt−1 to286

a new position θ̃t are drawn from a proposal distribution q(·) and accepted287

based with the Metropolis acceptance probability288

κ(θt−1, θ̃t) =

min(

π(θt−1)

π(θ̃t)

)

, if π(θ̃t) > 0

1, if π(θ̃t) = 0

(6)

13

where π(·) stands for the likelihood of a parameter set, which increases289

with decreasing Eq. 5. The candidate position is always accepted as θt if290

it has a higher likelihood, otherwise there still is a chance of moving there291

(otherwise, θt = θt−1). This has at least two consequences: (a) the chain292

can move out of local minima by chance and more importantly, (b) under293

certain requirements for the proposal, the chains will converge and sample294

only from a unique stationary distribution. This distribution is equal to the295

posterior distribution of the parameters, containing our uncertain knowledge296

about the optimal parameter region.297

The key point in designing an MCMC algorithm is choosing the proposal298

distribution q(θt−1, ·). Careless choice of the proposal will lead to slower con-299

vergence, which in our case of physical hydrological models translates into300

excessive computational demand. A major inefficiency in simple proposals is301

that they lack recognition of scale and orientation of the stationary distribu-302

tion. To this end, DREAM runs multiple Markov chains in parallel. Apart303

from the benefits for calculating convergence statistics (e.g. Gelman and Ru-304

bin, 1992), dealing with multimodal distributions or long-tailed distributions305

(see e.g. Laloy and Vrugt, 2011, and references therein), this allows for an306

elegant way of creating proposals that self-adjust to the scale and orientation307

of the target distribution (ter Braak, 2006): Jumps in each chain are gener-308

ated from the vector connecting two other chains.309

Formally, a candidate position for a jump in chain i is found as310

θ̃it = θit−1 + γ(

θjt−1 − θkt−1

)

+ ǫ , where i 6= j 6= k (7)

with an ergodicity term ǫ that is drawn from a symmetric distribution with311

variance small compared to the posterior distribution, and allows for some312

14

random parameter space investigation. This proposal can be combined with313

a self-adaptive randomized subspace sampling method to achieve a MCMC314

algorithm with desirable theoretical properties of maintaining detailed bal-315

ance and ergodicity (ter Braak and Vrugt, 2008; Vrugt et al., 2008a, 2009a).316

This principle is demonstrated on Fig. 7, which shows a slice through the317

parameter space for two parameters of the model in the optimization of the318

2009 period. A larger (or less negative) objective function, corresponding to319

the blue areas, is the posterior region. Two chains are shown at uniformly320

sampled positions of their evolution in parameter space. The first jumps321

(marked by the thick arrow) were still rather random, but then it is clearly322

visible that the jumps orient their direction and scale towards the region of323

higher likelihood, and end up sampling just from that region, the posterior324

distribution.325

Three modifications have been proposed that lead to the current variant326

(MT)DREAMZS. First, the ZS stands for sampling from an archive of past327

states. Sampling only from the current positions of chains as in the original328

DREAM requires at least d/2 to d chains, where d is the number of pa-329

rameters. This becomes very inefficient for problems with many parameters.330

Therefore, Vrugt et al. (2011) proposed keeping an archive of past states to331

generate candidate points. The resulting DREAMZS needs only d = 3 . . . 5332

chains and requires less function evaluations to converge.333

A second modification is the introduction of Snooker updates to increase334

proposal diversity (ter Braak and Vrugt, 2008), where the difference vector335

in Eq. 7 is projected unto a line connecting the current chain position to336

another chain.337

15

The third addition is a multi-try algorithm implemented by Laloy and Vrugt338

(2011) that incorporates a multiple-try Metropolis method (Liu et al., 2000)339

into DREAM that tests multiple candidate points mixing small and large340

jumps simultaneously. Candidate points are then selected with probability341

proportional to their likelihood and accepted with a Metropolis ratio. This342

approach leads to higher acceptance of candidate points and faster conver-343

gence.344

4.2. Implementation345

To take advantage of the potential of running in parallel multiple chains,346

and within each chain, multiple tries, (MT)DREAMZS is implemented to347

run on a computing cluster, running in our variant 3 chains in parallel, and 3348

candidate points are created simultaneously for each chain. Snooker updates349

are chosen over the DREAM parallel update with a probability of 10%.350

The build the objective function that measures the distance SS between351

modeled and measured values and is minimized by the optimization, we need352

to add the different measurement types in a way that gives equal weight to353

each series. To be able to compare all soil water measurements in one series,354

we calculate stored water volume by multiplying each soil moisture probe’s355

value with the volume it representsand restrict the calculation to a depth of356

90 cm. In calculating the error between measured and modeled storage, we357

take each volume’s difference first and then sum, to keep the local information358

intact, so we obtain359

SSStore =N∑

n=1

V∑

v=1

(yi,store,v − Yi,store,v)2 (8)

16

where the second sum runs over all volume elements 1 . . . V .360

We weigh the individual measurement types by the number of measurements361

and the measurement errors, resulting in objective function for the 2009362

series:363

O09 =SSStem

σ2err,Stem

+nSap

nStem

· SSSap

σ2err,Sap

+nStore

nStem

· SSStore

σ2err,Store

(9)

where the measurement errors σerr are heuristically set to 20% of the standard364

deviation of each respective measurement series, and nStem, nSap and nStorage365

and the number of measurements in the stem water potential, sap flux and366

storage time series.367

For the 2010 period, we have no stem water potential measurements available,368

therefore the above equation reduced to369

O10 =SSSap

σ2err,Sap

+nSap

nStore

· SSStore

σ2err,Store

(10)

5. Results and Discussion370

All the optimized parameters and their uncertainty are listed in Tables371

1 and 2 with the range of the uniform prior distribution and the 95% and372

50% confidence intervals along with the medians in the posterior distribu-373

tion. To help assess how much the parameters were constrained and if the374

same parameter ranges were found for the two different optimization scenar-375

ios, Fig. 8 shows a caterpillar plot of the parameters. The rainfall scaling376

parameter srain has been omitted since it was only optimized for the 2010377

scenario. The figure shows the prior ranges in each parameter scaled to the378

interval between 0 and 1. The orange bars are for 2009, and are in each case379

immediately followed by an orange bar for the same parameter in the 2010380

17

scenario. The width of the narrow bar indicates the 95 % confidence interval381

of the posterior, the thick part the 50 % confidence interval; and the whisker382

marks the median.383

((TODO: Discuss how much ranges were constrained, which parameters were384

not identified well and where the optimizations disagree. Needs better con-385

vergence.)) A lot of the parameters agree within the margin of error, but386

some of the parameters were found differently for the scenarios. Among these387

are α and n of the tree and the hydraulic conductivities of the soil. The val-388

ues for ssap are expected to be different, since they compensate for different389

sap flux amplitudes.390

5.1. Soil Water Characteristics391

We compare the optimized hydraulic parameters of the tree to data re-392

ported in the literature. A brief description of the literature included is393

presented in appendix 7.1. Most studies are concerned in establishing a rela-394

tion between relative hydraulic conductivity and water potential (or applied395

pressure), a so-called vulnerability curve. To facilitate comparison to these396

data, we use the optimized parameters to connect pressurehead and relative397

conductivity. To acknowledge the parameter uncertainty contained in the398

parameter posterior distributions that were determined by (MT)DREAMZS,399

we apply a bootstrap method by randomly drawing parameter combinations400

from the posterior distribution and displaying the area enclosing 95 % of the401

resulting vulnerability curves. The resulting curves are found in Figures 10402

and 11.403

Again, due to the much narrower bounds on the 2010 optimization, the un-404

certainty band is very narrow as well, but is mostly in agreement within the405

18

2009 band uncertainty. (???) The relative conductivity for the optimized pa-406

rameters generally decreases at lower pressures than in the curves reported407

in the literature. There is, however, good agreement to the data by Waring408

and Running (1978), but Domec and Gartner (2001) note that their method409

of cutting small flat discs of wood cuts open tracheids and leads to measure-410

ments neglecting the tension within the xylem stream, thus overestimating411

water capacitance, leading to underestimation of relative conductivity. Dis-412

crepancies, however, may also stem from the fact that the literature studies413

are conducted on branches or wood pieces only, and that we rather estimate414

effective hydraulic parameters for the complete tree ??evidence??.415

A second reason for the discrepancy might stem from correlation to other416

parameters, like the scaling parameters ssap, sET0 or soil hydraulic parame-417

ters. To test for possible influence through correlation to other parameters,418

we check the correlation matrix of the posterior distribution. We use the re-419

cently developed distance correlation (Szekely et al., 2007; Szekely and Rizzo,420

2009), which is a new measure of dependence between random vectors. It421

has the desirable property of being 0 only if both vectors are independent422

and was found to perform excellent across a range of applications. Figure 12423

shows the correlation between the tree hydraulic properties and all parame-424

ters in the optimization for 2009.425

Strong correlations are found between n and α. A strong correlation be-426

tween these parameters has been described before (see e.g. Huisman et al.,427

2010), and could plausibly be cause for some uncertainty about the relative428

conductivity (??as it determines the curve shape??). The strongest correla-429

tion, however, is found between the saturated hydraulic conductivity and the430

19

sapflux scaling factor. Since this factor only had to be included to account431

for uncertainty in quantiyfing the amplitude of sapflux, future studies with432

improved sapflux measurements should find different vulnerability curves if433

this correlation introduced an error into the optimization. Some data were434

also available on the soil water characteristics, and presented with the opti-435

mized retention curves in Figures 13 and 14, again with the 95 % confidence436

area of the optimized parameters. Here, we find good agreement within the437

uncertainty bounds both with the Waring and Running (1978) and Yoder438

(1984) studies for both years, although the 2010 uncertainty bounds again439

are much narrower.440

441

5.2. Modeled vs. measured data442

To evaluate how good the best set of parameters describes the measured443

data, we compare measured and modeled data in Figs. 15 and 16. For 2009,444

the potential evapotranspiration ET0 as determined from the flux tower mea-445

surements is shown in the top panel, and the site-specific potential evapo-446

transpiration ETp as scaled with the parameter sET0 in green. The three447

different measurement types are shown in the next panels. A very good448

agreement can be found for all of the three measurement series.449

The modeled sap flux can not completely capture the highest amplitudes at450

noon. However, as the dynamics of the time series are captured very well,451

and as the amplitude of the sap flux is scaled by the parameter ssap we must452

conclude that the optimization was not able to find a better find of the dy-453

namics with scaling leading to higher amplitudes. This, on the one hand, is454

encouraging since it vindicates our assumption that introducing scaling on455

20

the sap flux would be justifiable since catching the dynamics was the primary456

objective. On the other hand, this also indicates that there is some error in457

the model setup that does not allow for even better sap flux dynamics. This458

might be connected to the forcing and the scaling to the site, or to structural459

model error through simplification, e.g. the assumption of axial symmetry.460

An additional test was undertaken to check if the choice of sap flux scaling461

might have been influential. We chose a scaling model that included an ad-462

ditional intercept parameter, however this did not lead to a better fit of the463

amplitudes.464

The stem water potential measurements are all captured very well by their465

model counterparts, except one very low measurement that was taken around466

3 pm. This one exception might be connected to microconditions, like the467

branch getting more sunlight during this time and thus not being completely468

representative of the tree, it might also be connected to the same issues that469

lead the model to not fit the highest amplitudes in the sap flux quite right,470

or to the heterogeneities in the actual tree that are locally not captured by471

the effective hydraulic parameter we obtain from our optimization. To check472

for an effect of the measurement series weighting, we investigated a variant473

of the optimization with even larger weight on the stem water potential mea-474

surements. This lead to only marginally better modeling of the stem water475

potential, but considerably larger discrepancies in the two other modeled476

data types. It will remain for future investigation whether a series with stem477

water potential measurements over a larger time period would improve the478

optimization even further.479

Finally, the modeled storage shows very good agreement with the measured480

21

values, only somewhat overestimating the remaining water content in the last481

five days. This could be connected to the simple two layer model of the soil482

or to the root density distribution.483

484

5.2.1. 2010 Period485

For the second optimization scenario, the 2010 rainfall period, no stem486

water potential measurements were available. Figure 16 shows the evapo-487

transpiration, the measured and scaled rainfall, and the sap flux and storage488

time series. While the storage again is modeled well, there are larger dis-489

crepancies between modeled and measured sap flux, most prominently the490

amplitudes are severely underestimated.491

There could be a number of factors contributing. First, it is visible that the492

measured time series is of more questionable quality. We already mentioned493

that differences in amplitude compared to 2009 as motivation to introduce494

the scaling factor ssap. But the time series also contains a baseline value495

it doesn’t fall below, and that wasn’t present in the 2009 series. It is not496

very plausible that the tree actually kept a constant flux value during the497

night time under changing environmental conditions. This rather could be498

instrument degradation, as it is a well known effect in sap flux probes that499

the tree regrows around the wound inflicted by the probe installation and500

possibly cuts off the probe from sap flow paths. This would also explain the501

overall lower amplitudes in 2010 compared to 2009, however this is not com-502

pletely clear at this point. This baseline can not be captured by the model503

and throws off the optimization. Additionally, spurious sap flux measure-504

ment can be seen on day 277 and 278. These however do not seem to have505

22

any impact on the quality of the optimization. In addressing the baseline506

problem, experiments with a two parameter sap flux scaling with intercept507

as described above did not lead to a better fit. Future measurements in the508

project with newly installed sap flux sensors will provide better information509

about the complexities of the sap flux measurements and hopefully improve510

the optimization results.511

A second reason might be the spatial separation of rainfall measurements and512

tree. We somewhat addressed this with a rainfall scaling factor, but while513

this can cover the problem of the amount of intercepted rainfall and surface514

runoff, adjusting for the site is not readily possible with this approach. Fur-515

thermore, we are forced to distribute daily rainfall events as hourly input516

over the whole day, thereby manipulating the timing of the rainfall onset.517

For example, on day 279 there is still some sap flux and evaporation mea-518

sured even though a rainfall event happened probably later in the day. As519

the modeled storage immediately after rainfall events overestimates the wa-520

ter content, it can be concluded that the model somewhat oversimplifies the521

water infiltration due to rainfall due to these limitations.522

A third reason could be the missing stem water potential measurements.523

From the 2009 period it became clear that when giving about equal weight524

to each measurement type in the objective function, the agreement between525

all three modeled and measured series increased. The stem water potential526

holds valuable information to estimate the hydraulic parameters of the tree,527

thus not including them might provide insufficient information to find the528

best parameters.529

530

23

5.3. Vertical Water Fluxes531

The transport of water is governed by vertical flow processes, both in532

the soil and the tree. Figure 9 shows vertical fluxes averaged over segments533

of certain length and, for the tree, time periods. In the soil, fluxes are534

mostly downwards in 2009 with some upwards movement near the surface,535

and strongly downwards in 2010 due to dominating rainfall pattern. In both536

years, a peak of less downwards flow can be seen around -100 cm depth.537

This is the maximum of the root density distribution (see Figure 4), leading538

locally to water being transported into this region to replace water taken up539

by roots.540

In the tree, water is transported upwards into the canopy. Above 600 cm,541

water is extracted from sink terms along the tree nodes, and the vertical542

flux reaches a peak, that gradually goes down towards zero. The vertical543

fluxes in 2010 exhibit similar behavior as the 2009 ones, but with smaller544

absolute values, due to the smaller potential evapotranspiration. Vertical545

flux is clearly higher during the hours between 8 am and 4 pm, and the546

decrease after the peak is stronger.547

(...)??more depending on converged results???548

6. Summary and Outlook549

TEXT550

7. Appendix551

??Find out how to restart figure numbering after appendix environment552

so this doesn’t have do be a section.??553

24

7.1. Literature Data554

This section describes the literature data used for the comparison of the555

tree hydraulic functions in Figs. 10 to 14. Waring and Running (1978)556

analysed the sapwood of tall Douglas Firs from the Cascade Mountains of557

Oregon. They measured relative water content in the laboratory, and related558

them to water potential measurement and the relative change in conductivity559

that was estimated from measurements of transpiration. This data set was560

presented as relative conductivity against saturation, so we translate this561

data set into vulnerability curves using our best parameter sets.562

From the thesis of Yoder (1984), we included measurements on a White563

Fir. Twig samples were taken from young trees (10-13 years old) and a564

water retention curve was measured in the laboratory using a pressure bomb565

technique as established by (Tyree and Hammel, 1972). Measurements were566

taken at different times over year, and we include data from March, June567

and August.568

A number of branches of mature tree specimen were analysed using hydraulic569

conductivity and acoustic measurements by Cochard (1992b) to investigate570

vulnerability to air embolism and cavitation. From his data, we include571

vulnerability curve measurements on a Douglas Fir.572

Sperry and Ikeda (1997) compared air-injection and dehydration methods573

for determining the vulnerability curves of conifers. As both methods were574

found to obtain similar results, we restrict the data included to White Fir575

dehydration measurements on branch segments (10-20 cm long)576

A comparison of the vulnerability curves for outer and inner sapwood on577

young (1.0-1.5 m tall) and mature Douglas firs (41-45 m tall) was included578

25

(Domec and Gartner, 2001), as well as a vulnerability curves established579

through a centrifugal method as described by Li et al. (2008).580

To allow some comparison to other tree species, we also include newer results581

by Gonzalez-Benecke et al. (2010) and Johnson et al. (2011) that establish582

vulnerability curves on Red Maple, American Tulip Trees, Virginia Pines,583

Slash Pines and Longleaf Pines.584

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34

Figure 1: Site scheme. The six pits with soil moisture probes used for this study are

marked with blue circles.

Figure 2: Schematics of the model.

35

Figure 3: Scheme of water flow boundary processes.

Figure 4: Optimized root density distribution (2009 period) to a depth of 2.5 m.

36

P4 P3 P2 P1Pressure [cm]

0.0

0.2

0.4

0.6

0.8

1.0α(h

)

Figure 5: Water stress response function α(h) and characteristic pressure values.

Figure 6: Initial pressure head distribution in the soil for the 2009 period.

37

−800

−700

−600

−500

−400

−300

−200

−100

0.2 0.3 0.4 0.5 0.6 0.7 0.8

0.35

0.40

0.45

0.50

0.55

Kc

soil1

thet

aS

Figure 7: Example of a slice of the objective function for two parameters and the evolution

of two chains. Only 12 positions are shown for each chain, evenly spaced in time. The

thicker arrows marks the first jump.

38

0.0 0.2 0.4 0.6 0.8 1.0

h_ini,Tr (2010)h_ini,Tr (2009)

theta_S, Soi2 (2010)theta_S, Soi2 (2009)theta_S, Soi1 (2010)theta_S, Soi1 (2009)log k_S,Soi2 (2010)log k_S,Soi2 (2009)log k_S,Soi1 (2010)log k_S,Soi1 (2009)

log k_S,Tr (2010)log k_S,Tr (2009)

n_Tr (2010)n_Tr (2009)

alpha_Tr (2010)alpha_Tr (2009)

theta_S,Tr (2010)theta_S,Tr (2009)

P_3, Tr (2010)P_3, Tr (2009)P_2, Tr (2010)P_2, Tr (2009)P_2,Soi (2010)P_2,Soi (2009)

s_ET0 (2010)s_ET0 (2009)s_sap (2010)s_sap (2009)

p_m (2010)p_m (2009)z_m (2010)z_m (2009)p_z (2010)p_z (2009)p_r (2010)p_r (2009)

Figure 8: Caterpillar plot

39

−500 −400 −300 −200 −100 0

−1e

−03

−4e

−04

2e−

04

(a) Soil 2009

Depth [cm]

Ver

tical

Flu

x [c

m/d

]

−500 −400 −300 −200 −100 0−

2.0

−1.

00.

0

(b) Soil 2010

Depth [cm]V

ertic

al F

lux

[cm

/d]

0 500 1500 2500

−0.

20.

20.

6

(c) Tree 2009

Height [cm]

Ver

tical

Flu

x [l/

h]

MiddayOther Hours

0 500 1500 2500

−0.

20.

20.

6

(d) Tree 2010

Height [cm]

Ver

tical

Flu

x [l/

h]

MiddayOther Hours

Figure 9: Vertical fluxes averaged over segments of 50 cm (soil) or 300 cm (tree) length.

(c) and (d) in the lower row are segmented in time for hours between 8 am and 4 pm or

other hours. The segmented data was approximated by a smoothing spline.

40

0

20

40

60

80

100

-60000 -50000 -40000 -30000 -20000 -10000 0

Rel

ativ

e C

ondu

ctiv

ity [%

]

Pressure Head [cm]

Figure 10: Vulnerability curves 2009. Median of optimized posterior distribution as thick

green line, blue area gibves 95% confidence interval. Literature data: (×) White Fir

(Waring and Running, 1978), (N) Inner Sapwood, 1 m (H) Inner Sapwood, 35 m (�)

Outerwood Sap, 1 m (�) Outer Sapwood, 35 m - Douglas Fir (Domec and Gartner, 2001),

(×) White Fir, Dehydration (Sperry and Ikeda, 1997), (+) Red Maple, (N) American

tulip tree, (H) Virginia Pine (Johnson et al., 2011), (�) Slash Pine, (�) Longleaf Pine

(Gonzalez-Benecke et al., 2010), (+) Douglas Fir (Cochard, 1992b), (×) White Fir (Li

et al., 2008).

41

0

20

40

60

80

100

-60000 -50000 -40000 -30000 -20000 -10000 0

Rel

ativ

e C

ondu

ctiv

ity [%

]

Pressure Head [cm]

Figure 11: Vulnerability curves 2010. Median of optimized posterior distribution as thick

green line, blue area gibves 95% confidence interval. Literature data: (×) White Fir

(Waring and Running, 1978), (N) Inner Sapwood, 1 m (H) Inner Sapwood, 35 m (�)

Outerwood Sap, 1 m (�) Outer Sapwood, 35 m - Douglas Fir (Domec and Gartner, 2001),

(×) White Fir, Dehydration (Sperry and Ikeda, 1997), (+) Red Maple, (N) American

tulip tree, (H) Virginia Pine (Johnson et al., 2011), (�) Slash Pine, (�) Longleaf Pine

(Gonzalez-Benecke et al., 2010), (+) Douglas Fir (Cochard, 1992b), (×) White Fir (Li

et al., 2008).

42

Figure 12: Extract of the distance correlation matrix for the 2009 parameter posterior

distributions. Larger and more intensely red circle represent a large correlation.

43

0

20

40

60

80

100

120

140

0 0.2 0.4 0.6 0.8 1

Pre

ssur

e H

ead

[103 c

m]

Saturation

Figure 13: Soil Water Characteristics 2009. Median of optimized posterior distribution as

thick green line, blue area gives 95% confidence interval. Literature data: (�) White Fir

(Waring and Running, 1978), (+) White Fir, March (N) June (H) August (Yoder, 1984).

44

0

20

40

60

80

100

120

140

0 0.2 0.4 0.6 0.8 1

Pre

ssur

e H

ead

[103 c

m]

Saturation

Figure 14: Soil Water Characteristics 2010. Median of optimized posterior distribution as

thick green line, blue area gives 95% confidence interval. Literature data: (�) White Fir

(Waring and Running, 1978), (+) White Fir, March (N) June (H August (Yoder, 1984).

45

195200

205210

0.0 0.2 0.4 0.6 0.8 1.0

ET0 [cm/d]

195200

205210

0 5 10 15 20 25 30 35

Sap Flux [l/h]

195200

205210

−15000 −10000 −5000 0

Stem Pot. [cm]

195200

205210

7800000 8400000 9000000

Day in 2009

Storage [cm^3]

Figu

re15:

Forcin

g,modeled

andmeasu

reddata

for2009

optim

ization.

46

275 280 285 290 295 300

0.0

0.2

0.4

0.6

ET

0 [c

m/d

]

275 280 285 290 295 300

02

46

Pre

cipi

tatio

n [c

m/d

]

275 280 285 290 295 300

05

1015

Sap

Flu

x [l/

h]

275 280 285 290 295 3006.0e

+06

1.0e

+07

1.4e

+07

Day in 2010

Sto

rage

[cm

^3]

Figure 16: Forcing, modeled and measured data for 2010 optimization.

47

Table 1: Prior parameter range and posterior parameter 50 % and 95 % confidence interval for 2009 period.

Parameter Lower Upper 2.5 % 25 % Median 75 % 97.5 %

pr [-] −1 4 -0.9504 0.1047 0.7597 1.487 3.887

pz [-] .5 4 1.621 3.216 3.676 3.897 3.981

zm [cm] 0 100 1.522 76.78 88.44 96.23 99.4

rm [cm] 500 2500 557.9 730.3 1115 1584 2079

ssap [-] 0 5 0.1295 0.1793 0.3293 0.4174 0.5368

sET0 [-] 0.1 1 0.05691 0.07949 0.1958 0.2322 0.3257

P2, Soi [cm] −2500 −20 -2276 -1177 -798.8 -334 -64.44

P2, Tr [cm] −20000 −500 -19750 -15470 -5815 -3533 -719.1

P3, Tr [cm] −50000 −15000 -48850 -39260 -34560 -24370 -18550

θS, Tr [cm3/cm3] 0.4 0.7 0.4031 0.4557 0.5411 0.5944 0.6804

αTr [cm−1] 2.5 · 10−5 7 · 10−5 2.561 · 10−5 2.714 · 10−5 3.652 · 10−5 4.21 · 10−5 4.66 · 10−5

nTr [-] 2 15 4.186 4.873 5.588 7.484 10.87

log kS, Tr [cm/d] .75 3 0.7799 0.8986 1.3 1.387 1.556

log kS, Soi1 [cm/d] 1 3 1.007 1.062 1.114 1.171 1.286

log kS, Soi2 [cm/d] 1 3 1.004 1.022 1.049 1.087 1.325

θS, Soi1 [cm3/cm3] 0.3 0.6 0.5215 0.5768 0.5872 0.5958 0.5993

θS, Soi2 [cm3/cm3] 0.3 0.6 0.5071 0.5553 0.5746 0.5892 0.5998

hini, Tr [cm] −10000 −1000 -1619 -1264 -1133 -1050 -1002

48

Table 2: Prior parameter range and posterior parameter 50 % and 95 % confidence interval for 2010 period.

Parameter Lower Upper 2.5 % 25 % Median 75 % 97.5 %

pr [-] −1 4 -0.971 -0.4508 0.735 1.392 3.265

pz [-] .5 4 2.859 3.067 3.47 3.697 3.982

zm [cm] 0 100 30.53 64.41 72.4 94.29 99.87

rm [cm] 500 2500 708.7 1423 1672 1991 2483

ssap [-] 0 5 0.5416 0.5518 0.5883 0.5946 0.6154

sET0 [-] 0.1 1 0.2649 0.2701 0.2719 0.2742 0.285

srain [-] 0.1 1 0.1974 0.396 0.4407 0.512 0.6221

P2, Soi [cm] −2500 −20 -2451 -2355 -1878 -1656 -1471

P2, Tr [cm] −20000 −500 -17560 -16850 -16690 -16420 -15840

P3, Tr [cm] −50000 −15000 -18230 -1750 -17180 -17070 -16510

θS, Tr [cm3/cm3] 0.4 0.7 0.4004 0.4037 0.409 0.4138 0.4196

αTr [cm−1] 2.5 · 10−5 7 · 10−5 4.807 · 10−5 4.992 · 10−5 5.098 · 10−5 5.138 · 10−5 5.353 · 10−5

nTr [-] 2 15 14.67 14.83 14.88 14.92 14.98

log kS, Tr [cm/d] .75 3 0.7582 0.7614 0.7645 0.7752 0.7918

log kS, Soi1 [cm/d] 1 3 2.741 2.843 2.978 2.993 2.999

log kS, Soi2 [cm/d] 1 3 2.143 2.287 2.372 2.463 2.581

θS, Soi1 [cm3/cm3] 0.3 0.6 0.4785 0.5084 0.5224 0.5307 0.5623

θS, Soi2 [cm3/cm3] 0.3 0.6 0.4604 0.4735 0.4971 0.521 0.5586

hini, Tr [cm] −10000 −1000 -1048 -1032 -1022 -1016 -1010

49