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: [email protected] (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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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