Tree Physiology 27, 1093–1102
© 2007 Heron Publishing—Victoria, Canada
An alternative method to estimate zero flow temperature differences
for Granier’s thermal dissipation technique
CARLOS M. REGALADO1,2 and AXEL RITTER1
1
Dept. Suelos y Riegos, Instituto Canario de Investigaciones Agrarias (ICIA), Apdo. 60 La Laguna, 38200S/C Tenerife, Spain
2
Corresponding author ([email protected])
Received October 27, 2006; accepted December 20, 2006; published online May 1, 2007
Summary Calibration of the Granier thermal dissipation
technique for measuring stem sap flow in trees requires determination of the temperature difference (∆T ) between a heated
and an unheated probe when sap flow is zero (∆Tmax ). Classically, ∆Tmax has been estimated from the maximum predawn
∆T , assuming that sap flow is negligible at nighttime. However, because sap flow may continue during the night, the maximum predawn ∆T value may underestimate the true ∆Tmax. No
alternative method has yet been proposed to estimate ∆Tmax
when sap flow is non-zero at night. A sensitivity analysis is presented showing that errors in ∆Tmax may amplify through sap
flux density computations in Granier’s approach, such that
small amounts of undetected nighttime sap flow may lead to
large diurnal sap flux density errors, hence the need for a correct estimate of ∆Tmax. By rearranging Granier’s original formula, an optimization method to compute ∆Tmax from simultaneous measurements of diurnal ∆T and micrometeorological
variables, without assuming that sap flow is negligible at night,
is presented. Some illustrative examples are shown for sap flow
measurements carried out on individuals of Erica arborea L.,
which has needle-like leaves, and Myrica faya Ait., a broadleaf
species. We show that, although ∆Tmax values obtained by the
proposed method may be similar in some instances to the ∆Tmax
predicted at night, in general the values differ. The procedure
presented has the potential of being applied not only to
Granier’s method, but to other heat-based sap flow systems that
require a zero flow calibration, such as the Èermák et al. (1973)
heat balance method and the T-max heat pulse system of Green
et al. (2003).
Keywords: Granier method, nighttime sap flow, thermal dissipation probe, water uptake, wood heat conductivity.
Introduction
Among methods for estimating whole-tree water demand, the
use of heat dissipation and heat balance sensors are affordable
and relatively simple techniques (Wullschleger et al. 1998).
However, one drawback of heat-based sap flow systems, such
as the Granier thermal dissipation probe method (TDP), is that
they require calibration during a period of zero sap flow
(Granier 1985). A period of zero sap flow is usually assumed
to take place at nighttime, and a maximum temperature difference (∆Tmax) between a heated and a reference unheated probe
is recorded predawn. However, sap flow does not necessarily
cease in tree branches and roots at night; thus, estimates of sap
flux density based on such a principle may be incorrect. Sap
may continue to flow during the night because of water movement for new growth, water redistribution in roots (Sakuratani
et al. 1999, Burgess et al. 2000) and refilling of stem tissue water stores after prolonged drought (Lu et al. 1995, Goldstein et
al. 1998). Far from being an exception, nighttime sap flow is
widespread among species, and ranges from 5 to 30% of daily
water loss (Snyder et al. 2003, Ford et al. 2004, Ludwig et al.
2006).
Consequently, some authors have proposed that heat-based
sap flow systems should be calibrated when zero flow can be
ensured, for example, by severing the xylem conducting tissue
(Burgess et al. 2000) or based on the defoliation stage of the
tree (Do and Rocheteau 2002a). The former is a destructive
technique, and the latter may be valid only in deciduous trees.
Additionally, such methods assume that ∆Tmax is unique,
whereas both seasonal and short-term variations in maximum
thermal dissipation during the night have been reported. Although seasonal changes in ∆Tmax may be a consequence of
seasonal variations in wood water content, a sharp change or
drift in ∆Tmax may be a result of nighttime sap flow or water
stress (Do and Rocheteau 2002a, Lu et al. 2004). Furthermore,
because ∆Tmax is influenced by wood–probe contact effects,
thermal properties (heat conductivity) of the surrounding
wood and variability of the components of the sap flow probes,
it is a probe-specific black-box parameter, and therefore must
be computed independently for each sensor throughout the period of data collection.
The zero flux assumption is not restricted to TDP, but also
applies to the heat balance methods of Èermák et al. (1973)
and Sakuratani (1981), and the T-max heat pulse method of
Green et al. (2003). Consequently, some alternatives have
been proposed to avoid the problem of nighttime sap flow.
Èermák et al. (2004) acknowledged the problem of nighttime
flow in their trunk heat balance method, but claimed that errors
are small. By contrast, Sakuratani et al. (1999) reported a tech-
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REGALADO AND RITTER
nique to calculate the coefficient of the heat flow sensing element (the so-called sheath conductance) from constant power
heat balance gauges, without the need for zero flow during
night. Green et al. (2003) adopted a 7-day average predawn
value, to ensure the required zero sap flow condition necessary
to calculate the thermal diffusivity. Lu et al. (2004) have recently reviewed a double regression procedure to overcome
the problem of nighttime sap flow with the Granier TDP
method (see also Granier 1987). This approach has found limited use, but its accuracy has not been compared with alternative methods and it has not yet been demonstrated that the double regression procedure yields the true ∆Tmax if, for example,
there are seasonal changes in nighttime flow. The choice of a
required 7–10 day interval during which the true ∆Tmax is presumably achieved is also a drawback of such a regression
method, because of the subjectivity in the selected period
length and because it requires that, for an interpolation to be
possible, the true ∆Tmax must be found during several nights
within that interval.
Tatarinov et al. (2005) have shown that, for needle type
probes in the Granier system, the ratio of the temperature differences between the heated and the unheated probe inserted
in the tree trunk (∆T ) has high sensitivity to the heat conductivity of wood. Because ∆Tmax depends on wood heat conductivity (0.15 to 0.40 W m –1 K –1; Steinhagen 1977), changes in
wood water content are expected to produce significant variations in ∆Tmax. The scenario is further complicated because, as
we show in our study, Granier’s formula is sensitive to the
choice of ∆Tmax, so that undetected small amounts of sap flow
during the night may lead to large errors in sap flux density
computations.
The main objective of this paper is to describe an alternative
technique to compute ∆Tmax from concurrent measurements of
∆T and micrometeorological variables during daylight hours,
without the constraint of a nighttime zero flow. The method
makes use of optimization techniques and a restatement of
Granier's original formula. Some illustrative examples are
shown for sap flow measurements carried out on individuals of
a broadleaf species, Myrica faya Ait., and a coniferous tree
species, Erica arborea L. The measurements demonstrate that
the ∆Tmax value obtained from diurnal measurements may coincide in some instances with the ∆Tmax predicted at night, but
that in general the values differ.
Materials and methods
An alternative method for estimating ∆Tmax
An empirical relationship between sap flux density, v (kg m – 2
s – 1) and ∆T was originally obtained by Granier (1985) as:
v = αK β
(kg s – 1) is obtained from integration of Equation 1 across trunk
radial increments (dr) within the conducting sapwood radial
interval (rh, rx) as:
rx
QSF = 2π ∫ rv i dr
where vi is the sap flux density at radial depth i. Polynomial
functions are usually fitted to vi (r) measurements, such that an
analytical expression of the integral in Equation 2 may be obtained. This renders QSF as v times a multiplication factor,
which is a combination of the coefficients in the fitted polynomial and the radii (r; m) at the heartwood (rh) and the cambium, (rx ). The QSF values at the tree level may be extrapolated
to the stand by means of a botanical inventory and scaling with
an appropriate biometric parameter, such as diameter at breast
height (DBH) or basal area (Èermák et al. 2004).
Based on the relationship between potential evapotranspiration (ETp; mm) and actual transpiration estimated from
sap flow readings, we propose the following alternative
method for estimating ∆Tmax. Estimates of ETp may be derived
from micrometeorological measurements (Penman-Monteith,
eddy covariance, Bowen ratio) or a water balance approach
(e.g., weighing lysimeter). A proportionality law is thus derived from Equation 1, such that:
 ∆T

ETp = Bα max – 1

 ∆T
β
(3)
where the constant B in Equation 3 is considered as a dummy
parameter that comprises the multiplication factor resulting
from Equation 2, a biometric scaling index (see above) and an
appropriate unit conversion factor. It is reasonable to assume
that, on a daily basis, changes in soil water content are negligible. Hence, B also accounts for the weighting of soil water potential on ETp. This is not a drawback of the method proposed,
because different strategies are available to include the effect
of soil water status in the weighting of ETp (Stewart 1988,
Noilhan and Planton 1989, Lhomme et al. 1998, Hübener
2004). In our case, water content measurements were available
for the 15- and 30-cm depths but not for the whole soil profile
(1.2 m) affected by the root system, and so this last possibility
was not considered.
At first glance, optimization of Equation 3 may yield the
values of B and ∆Tmax from time measurements of ∆T and ETp.
Disappointingly, rearranging the terms in Equation 3 as:
ET βP = ( αB ) ∆Tmax
–1
(1)
where α = 0.119 kg m – 2 s –1 and β = 1.231 are empirical parameters, and K is the dimensionless flow index given by K =
∆Tmax / ∆T – 1 (for ∆Tmax ≥ ∆T ), such that, for zero sap flow periods, ∆T = ∆Tmax. Volumetric flux (QSF) or tree transpiration
(2)
rh
β –1
β –1
1
– ( αB )
∆T
(4)
shows that B and ∆Tmax are highly correlated. Hence, the optimization of parameters in Equation 4 cannot guarantee that the
fitted B and ∆Tmax are correctly identified. Although transforming Equation 3 by scaling variables between minimum
and maximum values as:
TREE PHYSIOLOGY VOLUME 27, 2007
ZERO FLOW ∆T ESTIMATION FOR GRANIER TDP
ETp – ETp,max
ETp,min – ETp,max
K β – K βmax
= ETp* = Bα β
=v*
K min – K βmax
(5)
reduces the parameter correlation, it does not altogether eliminate it (results not shown). However, Equation 4 is the equa–1
tion of a straight line with slope m = ( αB ) β ∆Tmax . For conve–1
nience, we have written ∆T as the dependent variable and
–1
6. Thus, a plot
ET βp as the independent variable in Equation
–1
of time measurements of ∆T –1 versus ET βp should render a
straight line, which crosses the OY axis at ∆T –1
max :
∆T – 1 =
–1
1
–1
ET βp + ∆Tmax
m
(6)
It is convenient to use only diurnal values for such a plot, because high uncertainty is expected in nighttime sap flow data
and also because nighttime values would contribute a scattered
data cloud near the origin, thus biasing the fitting. Together
with diurnal values, the maximum ∆T at night can be included
for estimating ∆Tmax by linear regression. In this case, a robust
fitting is suggested to down-weight outliers. The method proposed does not rely on a correct value of Granier’s empirical
–1
parameter α, because it is already included in (αB)–β . Also, if
a correct value of β were required, one could try to find a suit-
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able power transformation that linearizes the ∆T – 1 versus
–1
ET βp data, such that a suitable parameter α may be found for
the species investigated.
A further hypothesis is that Equation 6 will not be fulfilled
at all times during the day, but that tree transpiration and ETp
desynchronize at some stages. Lag effects between sap flow
and, for example, solar radiation or vapor pressure deficit have
been reported previously. Time lags associated with long-distance water transport in the stem and the use of stored water in
the sapwood result in asynchrony between basal stem flow and
canopy transpiration (Ford et al. 2004 and references therein).
Hence, before the linear fitting based on Equation 6 as de–1
scribed above, those (∆T – 1, ET βp ) data points that satisfy
(with a specified tolerance) the proportionality law introduced
in Equation 3 are selected by inverse optimization of Equation 5. In our case, this step was performed with a trust-region
algorithm (Conn et al. 2000), which is a nonlinear least square
method that is more efficient than the Levenberg Marquardt
optimization algorithm and others that are commonly used.
The method is summarized in Figure 1.
Description of the Granier sap flow system used
Measuring sap flow by the Granier TDP method (1985, 1987)
is based on the implantation of two cylindrical needle-like
Figure 1. Sketch summarizing the
steps carried out in the method
proposed to estimate ∆Tmax. Abbreviations: ETP* , transformed diurnal potential evapotranspiration;
v*, transformed diurnal sap flux
density; ∆T, temperature difference between a heated and an unheated probe; and β, dimensionless empirical parameter = 1.231.
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REGALADO AND RITTER
probes (L = 20 mm, ∅ = 2 mm; Up GmbH, Cottbus, Germany). The upper probe is continuously heated with a 34.5 Ω
(120 mA) resistor, whereas the lower probe is unheated and the
resulting temperature difference is measured with copper constantan (T-type) thermocouples (Seebeck coefficient 40µV
°C –1; relative accuracy 0.1 °C). A parallel resistor is added inside the plug to reduce the current flowing through the sensor
to 80 mA. Several centimeters of the heating wire are outside
the heating zone, thus the heating resistance must be reduced
by about 5.5 Ω, such that the heating power results in
(0.080A)2(34.5 – 5.5) Ω = 0.186 W. Because of variation in
heating wire and heat production, the real heating power is
2–10% higher than the claimed value of 186 mW (Up GmbH,
personal communication), but still close to the original value
in Granier's design (200 mW). Granier (1985) found that empirical coefficients in Equation 1 may depend on the heating
power used. A calibration is unavailable from Up GmbH for
their SFS-2 Granier-type sap flow system; however, Do and
Rocheteau (2002b) checked the validity of Granier original
parameter values α and β for Up GmbH sap flow sensors in
Plexiglas sawdust columns (see Figure 3 in Do and Rocheteau
2002b). We did not carry out a specific calibration for the sap
flow system in Erica arborea and Myrica faya, because it is
not required for the method proposed.
Field installation of the sap flow system
To illustrate the alternative method, we have applied it to measurements obtained with Erica arborea and Myrica faya trees
located in a plot at 1270 m a.s.l. in a 43.7-ha watershed in the
Garajonay National Park, La Gomera, Canary Islands (Ritter
et al. 2005). The forest vegetation present in the plot is locally
known as fayal-brezal, which is drought-adapted vegetation
usually located at hill crests and on upper hill slopes (Golubic
2001). Erica arborea and M. faya tree species are characterized by an active outer ring of conducting xylem, whereas half
of the inner stem is nonconducting (Jiménez et al. 2000). On
the north side of four 10-m-high trees, probe couples were inserted radially at 1.3-m height into the outer 20 mm of the stem
(excluding the bark) with a 10-cm vertical distance between
probes. After insertion, the drilled holes were sealed with insulating silicone, and the trunk measuring area protected from
solar radiation with an aluminum deflector. Probes were connected to a Combilog data logger (Up GmbH, Cottbus,
Germany) and signals recorded every 15 min.
Potential evapotranspiration
Estimates of ETp were obtained from micrometeorological
readings by the Penman-Monteith approach (Allen et al. 1998,
2006). Temperature and relative humidity (HMP45C thermohygrometer, Campbell Scientific Ltd., Lougborough, U.K.),
global radiation (SKS 1110 pyranometer, Skye Instruments
Ltd., Powys, U.K.) and wind velocity (A100R anemometer,
Campbell Scientific Ltd.) and direction (W200P wind vane,
Campbell Scientific Ltd.) data were collected with a 3-min
sampling frequency, and 15-min mean values were stored with
a Up GmbH Combilog data logger. Sensors were installed
above the stand on a 15-m-high scaffolding tower. Heights for
wind speed measurements, and for temperature and humidity
measurements were 13 and 12 m, respectively. Additionally,
two Time Domain Reflectometry (TDR) probes (Trime-EZ,
Imko GmbH, Ettlingen, Germany) were inserted at 0.15- and
0.30-m soil depths to monitor soil water content in the Ah horizon. The soil profile at the experimental site consists of a
4.5 cm leaf litter layer, followed by a 3-cm-deep hydrophobic
organic horizon and a 1.2 m loam to silt-loam mineral horizon
enriched with organic matter and exhibiting roots down to
1.15-m depth (Thissen 2001). A TDR soil-specific calibration
was used following Regalado et al. (2006).
The ETp values were computed with the following parameters. The albedo was taken as 0.14 (Aschan 1998), which is
close to the value of 0.11 proposed by Matthews (1984) for evergreen subtropical forests. The active leaf area index was calculated from Golubic (2001) and reduced by a shelter factor of
1.25 (Dingman 2002). The aerodynamic resistance was computed following Allen et al. (1998) with the expression derived
from turbulent transfers and assuming a logarithmic wind profile. The canopy surface resistance (rc) was considered to vary
with stomatal response to environmental factors. According to
Jarvis (1976) and Stewart (1988), leaf stomatal resistance
(rs,min ≤ rs ≤ rs,max) can be estimated as the product of the minimum resistance, rs,min (i.e., maximum conductance), corresponding to optimal conditions, and the various stress functions that vary with temperature, relative humidity, global radiation and leaf water potential (and thus also soil water potential). Following Lhomme et al. (1998), to compute the
potential evapotranspiration one may consider a surface resistance that depends on temperature and radiation, such that:
rc =
=
r
rs
= s,min
LAI act LAI act
rs,min ( k + R )

2
k 
LAI act  1 – 0.0016(25 – T )  R  1 +

Rmax 

(7)
, R>0
where LAIact is active leaf area index, T is air temperature (°C),
R is global solar radiation with Rmax= 1300 Wm – 2 and k is a
shape parameter. A value of k = 55 was derived by comparing
the Dickinson (1984) versus Stewart (1988) approach. A value
of rs,min = 176 s m – 1 was taken from the maximum stomatal
conductance reported by Tognetti et al. (2000) for Erica
arborea, and for Myrica faya, rs,min = 91 s m – 1 was used
(González-Rodríguez 1998). For R = 0, leaf stomatal resistance equals rs,max = 4100 s m – 1.
Software development
The method was implemented in a user-friendly computer program FITDTMAX, which performs the steps illustrated in
Figure 1. FITDTMAX uses the Penman-Monteith method and
Granier’s formula to calculate ∆Tmax from simultaneous measurements of micrometeorological variables and ∆T data ob-
TREE PHYSIOLOGY VOLUME 27, 2007
ZERO FLOW ∆T ESTIMATION FOR GRANIER TDP
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tained with thermal dissipation sap flow probes. The software
requires air temperature, relative humidity, solar radiation,
wind speed, ∆T time series, the maximum ∆T peak at nighttime (∆Tnight) and the parameters for the Penman-Monteith approach. A numerical and graphical output is provided for the
comparison between the computed ∆Tmax and ∆Tnight. FITDTMAX is available at http://www.icia.es/gh/software.html.
Results and discussion
Sensitivity analysis
To quantify how uncertainties in the prediction of the “true”
∆Tmax, due to either measurement errors or as a consequence of
an undetected nighttime sap flow, may affect sap flow computations with Granier’s formula (Equation 1), a sensitivity analysis was carried out. First, by inverting Equation 1 such that
∆Tmax = ∆Tmax(v):
  v β

∆Tmax = ∆T    + 1
 α



–1
(8)
we can investigate how changes in v may affect the determination of ∆Tmax. Figure 2A shows how small values of undetected
sap flow at night lead to significant changes in ∆Tmax. Thus, for
example, a low night sap flow of 0.007 kg m – 2 s – 1 yields a
10% change in ∆Tmax. Second, Figure 2B shows the results of a
sensitivity analysis for Granier’s formula (Equation 1) to variations in ∆Tmax for different flow indices (K ). For the sensitivity analysis, ∆Tmax is varied from 10 to –10% in ±1% increments, and the relative change in sap flux density (the difference between v and its initial value) is computed with Equation 1 for different values of K. Sensitivity curves are not
strictly symmetric but almost so. In general, errors in the assumed nighttime flow are amplified such that an undetected
nighttime sap flow of 0.007 kg m – 2 s –1 (–10% change in
∆Tmax) leads to relative changes of, for example, 0.016 kg m – 2
s –1 in sap flow values when K = 0.4. As K increases, variation
in ∆Tmax has a greater effect on the computed sap flux density.
For example, a 10% variation in ∆Tmax can lead to an underestimation of 0.041 kg m – 2 s –1 for large flows (K = 1.5) (Figure
2B). Additionally, as sap flow increases, the precision of
Equation 1 decreases (Tatarinov et al. 2005). Errors correspond to each single measurement, and are cumulative across
the day and along the measurement period. For example, assuming a mean hourly K = 0.4, an undetected nighttime sap
flow 0.007 kg m – 2 s –1 would lead to an hourly diurnal underestimation of about 61 dm3 m – 2 of conductive xylem. For low
K values, absolute errors are small, although in relative terms,
a 10% variation in ∆Tmax yields a change of about 1800% in v
for K = 0.05 (results not shown). Diurnal sap flow computations are not exempt from error even in the case of zero nighttime sap flow. Reported relative accuracy of copper-constantan (T-type) thermocouples used in TDP is generally 0.1 °C,
i.e., for ∆T = 10 °C, a 1% error may be assumed for the peak
nighttime ∆Tmax. Accordingly, this implies relative errors of
Figure 2. (A) Curve showing predicted change (%) in the temperature
difference between a heated and an unheated probe when sap flow is
zero (∆Tmax ) as a consequence of undetected low nighttime sap flow.
(B) Sensitivity analysis of sap flux density (ν) to variations in ∆Tmax
within the interval (–10%, 10%), for different flow indices (K ). In
those cases where the parameter combination yielded ν < 0, no sensitivity coefficient is shown.
6 × 10 – 4– 40 × 10 – 4 kg m – 2 s – 1 in v point measurements for K
values ranging from 0.05 to 1.5 (Figure 2B).
Practical application of the method
We applied the method to measurements made on trees of
Erica arborea and Myrica faya under different soil water content conditions (Table 1). For this purpose, the proposed
method was implemented in a computer program to carry out
data handling and computations automatically. Figures 3 and 4
show some illustrative examples for E. arborea and M. faya,
respectively. The transformed diurnal (> 10 W m – 2) ETp and v
dataset (ETp* and v*, respectively; see Equation 5) follow the
usual bell-shaped trend, rising from low morning values to a
plateau and then decreasing after midday to near zero. Data
pairs where the optimization procedure detected that the proportionality law (Equation 5) was satisfied, with a prescribed
0.05 tolerance, are marked with open circles. These were the
only data considered for the regression (Equation 6) (Figures 3
and 4, right-hand-side graphs). From these data, we discarded
low early morning and late afternoon values (those below the
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Table 1. Soil water content measured in the Ah horizon (θAh ), estimates of ∆Tmax – ∆Tnight, where ∆Tmax is the temperature difference between a
heated and an unheated probe when sap flow is zero, and ∆Tnight is the difference at night, and error in both sap flux density (RMSE) and daily totals of volumetric sap flow (QSF; in absolute and relative terms) for individuals of Erica arborea and Myrica faya tree species.
Species
θAh
(m3 m – 3)
∆Tmax – ∆Tnight
(°C)
RMSE
(kg m – 2 s – 1)
Error in QSF
(kg day – 1)
Relative QSF
error (%)
E. arborea
E. arborea
E. arborea
M. faya
M. faya
M. faya
0.508
0.367
0.222
0.472
0.295
0.216
0.44
0.53
0.02
0.10
0.37
0.34
0.0028
0.0036
0.0001
0.0006
0.0022
0.0016
0.940
1.877
0.035
0.251
1.733
1.080
27
22
1
8
17
27
dashed line in the left-hand-side graphs of Figures 3 and 4;
< 0.10) from the linear plot, because we expected larger uncertainty in those cases (Becker 1998, Perämäki et al. 2001,
Green et al. 2003). By contrast, the maximum ∆T at night
(∆Tnight ) was included (marked with a star in Figures 3 and 4,
right-hand-side graphs) and a robust fitting was carried out to
dampen its contribution. In general, goodness-of-fit was high
( r 2 ≥ 0.98, P < 0.01).
Figure 3 (lower panel) shows an example where ∆Tnight and
the ∆Tmax predicted by the regression method (䊐) agree
(17.3 °C). This example validates the potential of the proposed
method to predict ∆Tmax from the diurnal ∆T trend. However,
in general, ∆Tnight and ∆Tmax are different (Figures 3 and 4). In
some cases, differences may seem negligible but, as was earlier shown, a departure as low as 1–2% from the “true” ∆Tmax
may have a significant effect on cumulative daily sap flow (Table 1). The question now arises as to which of these estimates,
the ∆Tnight or the ∆Tmax obtained from Equation 6, is more reliable. Notice that the ∆Tnight is determined from a peak ∆T maximum, thus if nightly sap flow is present this would be undetected, and consequently ∆Tmax would be erroneous. Additionally, nighttime records are usually noisier, with a lower signal
to noise ratio, and thus the ∆Tnight will be more prone to errors,
especially if determined as a single maximum ∆T value and
not as a nighttime mean. By contrast, the ∆Tmax predicted from
the proposed method results from the linear trend of ∆T – 1 ver–1
sus ET βp optimized during daylight hours. In the optimization
procedure, the predicted ∆Tmax is unaffected by the presumed
nighttime sap flow, and the quality of the probe couple signal
is higher, with minimal error. Two further reasons support the
validity of the proposed technique. First, the ∆Tmax is the consequence of a linear trend detected during daylight hours. That
this trend will extend back to the predawn hours is, therefore,
to be expected. This is in a sense what the ∆Tmax computed as a
peak night value also assumes, i.e., that the wood thermal conditions prevailing at predawn apply during the following daylight hours. Second, if the wood thermal properties change on
a daily (or even shorter) time scale, it would then be more accurate to compute ∆Tmax from diurnal rather than nighttime ∆T
data, because the contribution to cumulative daily transpiration comes mainly from daylight sap flow and, therefore, the
daylight trend will be more representative of diurnal wood
thermal conditions.
Application to other heat-based sap flow systems
Although originally developed for Granier’s formula, the
method described here could be applied to other heat-based
sap flow systems requiring a zero flow calibration. For the
Èermák et al. (1973) heat balance method, for example, instead of Equation 6 we would have:
∆T –1 = mETp + b
(9)
where m = cwxh /BP and b = cwxhκ/P for known values of cw
(specific heat of water), xh (width of the heated volume) and P
(power input). Thus, the constant κ, usually estimated under
zero flow conditions, may be computed from the OY crossing
point of the line (Equation 9) fitted to the diurnal ETp, T – 1 data
pairs, i.e., κ = bP/cwxh. Granier’s formula and the Èermák et al.
(1973) model are both of the hyperbolic type and fitted closely
simulated data for needle, plate and volume heating configurations (Tatarinov et al. 2005). This means that the generality of
the proposed method is promising.
When the proposed approach is applied to the T-max heat
pulse method of Green et al. (2003), instead of Equation 6 we
would obtain the following quadratic dependence in the ET 2p ,
tM – 1 plane:
–2
–1
+ bt M
ET 2p = at M
(10)
with tM being the maximum temperature rise recorded by a
sensor located a distance xD downstream from a line heater.
From the parabola’s OX crossing point –b/a = 4κx D–1, one may
determine the required calibrating thermal diffusivity, κ.
Conclusions
An optimization procedure has been developed that permits
computation of ∆Tmax for Granier’s thermal dissipation technique, without the restriction of zero nighttime sap flow. A
slight modification of the proposed approach would be sufficient for its application in other heat-based sap flow systems,
which require a zero sap flow point calibration. From a practical point of view, data logger storage capacity and energy savings will be greatly increased with this method because, in
principle, only daylight hour data are necessary. This is impor-
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ZERO FLOW ∆T ESTIMATION FOR GRANIER TDP
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Figure 3. Example of the difference between a heated and an unheated probe when sap flow is zero (∆Tmax ) estimated in Erica arborea. Left panel:
diurnal transformed potential evapotranspiration (䊉, ETp* ) and sap velocity (䊐, v* ) rendered by the optimization process. Data selected for the
linear fitting are circled; dashed horizontal line delimits discarded early morning and late afternoon values (< 0.10). Right panel: robust linear fitting yielding ∆Tmax (䊐). The ∆Tnight (∗) indicates maximum ∆T at nighttime.
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Figure 4. Example of the difference between a heated and an unheated probe when sap flow is zero (∆Tmax) estimated in Myrica faya. Left panel:
diurnal transformed potential evapotranspiration (䊉, ETp* ) and sap velocity (䊐, v* ) rendered by the optimization process. Data selected for the
linear fitting are circled; dashed horizontal line delimits discarded early morning and late afternoon values (< 0.10). Right panel: robust linear fitting yielding Tmax (䊐). The ∆Tnight (∗) indicates maximum ∆T at nighttime.
TREE PHYSIOLOGY VOLUME 27, 2007
ZERO FLOW ∆T ESTIMATION FOR GRANIER TDP
tant where remote systems demand a high power input. Further work is aimed to optimize the data sampling frequency
necessary to detect possible changes in ∆Tmax during the day.
Acknowledgments
The authors thank Dr. Guido Aschan (on leave from University
Duisburg-Essen) for the installation of sap flow sensors and Thomas
Keller (UP GmbH) for the setup of field instrumentation. Support
from the Garajonay National Park is also acknowledged. This work
was financed with funds of the INIA-Programa Nacional de Recursos
y Tecnologías Agroalimentarias (Project RTA2005-228).
References
Allen, R.G., L.S. Pereira, D. Raes and M. Smith. 1998. Crop evapotranspiration: guidelines for computing crop water requirements.
Irrigation and Drainage Paper 56. Food and Agriculture Organization of the United Nations, Rome, 300 p.
Allen, R.G., W.O. Pruitt, J.L. Wright et al. 2006. A recommendation
on standardized surface resistance for hourly calculation of reference ETo by the FAO56 Penman-Monteith method. Agric. Water
Manage. 81:1–22.
Aschan, G. 1998. Mikroklima, Energiebilanz und Wasserhaushalt
von tropischen und extratropischen Wäldern. Edition Wissenschaft; Reihe Biologie, Bd. 159, Tectum-Verl., Marburg. In German.
Becker, P. 1998. Limitations of a compensation heat pulse velocity
system at low sap flow: implications for measurements at night and
in shaded trees. Tree Physiol. 18:177–184.
Burgess, S.S.O., M.A. Adams and T.M. Bleby. 2000. Measurements
of sap flow in roots of woody plants: a commentary. Tree Physiol.
20:909–913.
Èermák J., M. Deml and J.M. Penka. 1973. A new method of sap flow
rate determination in trees. Biol. Plant. 15:171–178.
Èermák, J., J. Kuèera and N. Nadezhdina. 2004. Sap flow measurements with some thermodynamic methods, flow integration within
trees and scaling up from sample trees to entire forest stands. Trees
18:529–546.
Conn, A.R., N.I.M. Gould and P.L. Toint. 2000. Trust-region methods. MPS-SIAM Series on optimization. SIAM, Philadelphia.
959 p.
Dickinson, R.E. 1984. Modeling evapotranspiration for three dimensional global climate models. Geophys. Monogr. 29:58–72.
Dingman, L. 2002. Physical hydrology. Prentice Hall, Upper Saddle
River, New Jersey, 656 p.
Do, F., and A. Rocheteau. 2002a. Influence of natural temperature
gradients on measurements of xylem sap flow with thermal dissipation probes. 1. Field observations and possible remedies. Tree
Physiol. 22:641–648.
Do, F. and A. Rocheteau. 2002b. Influence of natural temperature gradients on measurements of xylem sap flow with thermal dissipation
probes. 2. Advantages and calibration of a noncontinuous heating
system. Tree Physiol. 22:649–654.
Ford, C.R., C.E. Goranson, R.J. Mitchell, R.E. Will and R.O. Teskey.
2004. Diurnal and seasonal variability in the radial distribution of
sap flow: predicting total stem flow in Pinus taeda trees. Tree
Physiol. 24:951–960.
Goldstein, G.G, J.L. Andrade, F.C. Meinzer, N.M. Holbrook,
J. Cavelier, P. Jackson and A. Celis. 1998. Stem water storage and
diurnal patterns of water use in tropical forest canopy trees. Plant
Cell Environ. 21:397–406.
1101
Golubic, I. 2001. Vegetationskundliche Analyse von Lorbeerwaldund ähnlichen Beständen im Rahmen einer Untersuchung des
Landschftswasserhaushalts im Garajonay National Park La
Gomera (Kanarische Inseln). Diploma Thesis, Essen University
GH, Germany.
González-Rodríguez, A.M. 1998. Caracterización fotosintética de árboles de la laurisilva canaria (Laurus azorica, Persea indica y Myrica faya). Ph.D. Diss., University of La Laguna, Spain. In Spanish.
Granier, A. 1985. Une nouvelle méthode pour la mesure des flux de
sève dans le tronc des arbres. Ann. Sci. For. 42:193–200.
Granier, A. 1987. Evaluation of transpiration in a Douglas-fir stand by
means of sap flow measurements. Tree Physiol. 3:309–319.
Green, S., B. Clothier and B. Jardine. 2003. Theory and practical application of heat pulse to measure sap flow. Agron. J.
95:1371–1379.
Hübener, H., M. Schmidt, M. Sogalla and M. Kerschgens. 2004. Simulating evapotranspiration in a semi-arid environment. Theor.
Appl. Clim. 80:153–167.
Jarvis, P.G. 1976. The interpretation of leaf water potential and
stomatal conductance found in canopies in the field. Phil. Trans.
R. Soc. London Ser. B Biol. Sci. 273:593–610.
Jiménez, M.S., N. Nadezhdina, J. Èermák and D. Morales. 2000. Radial variation in sap flow in five laurel forest tree species in
Tenerife, Canary Islands. Tree Physiol. 20:1149–1156.
Lhomme, J.-P., E. Elguero, A. Chehbouni and G. Boulet. 1998.
Stomatal control of transpiration: examination of Monteith’s formulation of canopy resistance. Water Resour. Res. 34:2301–2308.
Lu, P., P. Biron, N. Breda and A. Granier. 1995. Water relations of
Norway Spruce (Picea abies (L.) Karst) under soil drought in the
Vosges mountains: water potential, stomatal conductance and transpiration. Ann. Sci. For. 52:117–129.
Lu, P., L. Urban and P. Zhao. 2004. Granier’s thermal dissipation
probe (TDP) method for measuring sap flow in trees: theory and
practice. Acta Bot. Sin. 46:631–646.
Ludwig, F., R.A. Jewitt and L.A. Donovan. 2006. Nutrient and water
addition effects on day- and nighttime conductance and transpiration in a C3 desert annual. Oecologia. DOI 10.1007/s00442006-0367-6.
Matthews, E. 1984. Vegetation, land-use and seasonal albedo data
sets. In Global Change Data Base Africa Documentation. Appendix D. NOAA/NGDC.
Noilhan, J. and S. Planton. 1989. A simple parameterization of land
surface processes for meteorological models. Monthly Weather
Rev. 117:536–549.
Perämäki, M., T. Vesala and E. Nikinmaa. 2001. Analysing the applicability of the heat balance method for estimating sap flow in boreal forest conditions. Boreal Environ. Res. 6:29–43.
Regalado, C.M., A. Ritter and R. Becker. 2006. Comments on “Monitoring soil water content profiles with a commercial TDR system:
comparative field tests and laboratory calibration.” Vadose Zone J.
5:1067–1068.
Ritter, A., C.M. Regalado, G. Aschan and L.A. Gómez. 2005. Contribución hídrica de la captación de niebla al balance de un bosque de
laurisilva en el Parque Nacional de Garajonay. In VII Jornadas de
Investigación en la Zona no Saturada del Suelo. Eds. J. Samper
Calvete and A. Paz González. Universidad da Coruña, pp 351–358.
In Spanish with english abstract. Available at: www.zonanosaturada.com/publics/ZNS05/area_5/05.pdf.
Sakuratani, T. 1981. A heat balance method for measuring water flux
in the stem of intact plants. J. Agric. Meteorol. 37:9–17.
Sakuratani T., T. Aoe and H. Higuchi. 1999. Reverse flow in roots of
Sesbania rostrata measured using the constant power heat method.
Plant Cell Environ. 22:1153–1160.
TREE PHYSIOLOGY ONLINE at http://heronpublishing.com
1102
REGALADO AND RITTER
Snyder, K.A., J.H. Richards and L.A. Donovan. 2003. Nighttime conductance in C3 and C4 species: do plant lose water at night? J. Exp.
Bot. 54:861–865.
Steinhagen, H.P. 1977. Thermal conductive properties of wood, green
or dry, from –40 to +100 °C: a literature review. USDA Forest Service Gen. Tech. Rep. FPL-09, Madison, WI, 10 p.
Stewart, J.B. 1988. Modeling surface conductance of pine forest.
Agric. For. Meteorol. 43:19–37.
Tatarinov, F.A., J. Kuèera and E. Cienciala. 2005. The analysis of
physical background of tree sap flow measurement based on thermal methods. Meas. Sci. Technol. 16:1157–1169.
Thissen, B. 2001. Untersuchung zur Abschätzung des Einfluss des
Bodens auf die Wasserbilanz von Teilflächen des Garajonay
Nationalparks, La Gomera, Spanien. Diploma Thesis, Essen University GH, Germany.
Tognetti, R., A. Minnocci, J. Peñuelas, A. Raschi and M.B. Jones.
2000. Comparative field water relations of three Mediterranean
shrub species co-occurring at a natural CO2 vent. J. Exp. Bot.
51:1135–1146.
Wullschleger, S.T., F.C. Meinzer and R.A. Vertessy. 1998. A review
of whole-plant water use studies in trees. Tree Physiol. 18:
499–512.
TREE PHYSIOLOGY VOLUME 27, 2007