Introduction

Plant stems are the pathways of soil water to the leaves for transpiration (Tyree and Sperry 1988). Measuring sap flow in stems and up-scaling it to plant transpiration can be conducted with several different sap flow methods such as the stem heat balance method, the heat pulse method or the thermal dissipation method (Smith and Allen 1996). Among these methods, the thermal dissipation probe (TDP) method (Granier 1985) is the most widely used one. Its advantages include its relatively low cost as well as relatively easy sensor construction and installation (Lu et al. 2004). The empirical TDP formula for the calculation of sap flux density (J s, g m−2 s−1) was first put forward by Granier (1985); J s is expressed as a function of the temperature difference (ΔT) between a heating probe and a reference probe: \({J_{\text{s}}}=119 \times {(\Delta {T_{{\text{max}}}}/\Delta T - 1)^{1.231}}\), where ΔT max is the ΔT under zero flow condition, which is commonly substituted by the diurnal nighttime maximum ΔT (Granier 1987).

As Granier’s formula was derived from an empirical relationship of three tree species (Pseudotsuga menziesii, Pinus nigra and Quercus pedunculate; Granier 1985) rather than being based on wood physical properties (Wullschleger et al. 2011), the TDP method has been reported to substantially over- or underestimate J s in various studies (Clearwater et al. 1999; Steppe et al. 2010; Bush et al. 2010). Potential reasons for observed divergences include non-uniform sap flow along the sensor (Clearwater et al. 1999), lacking compensation for the ‘wound effect’ (Wullschleger et al. 2011) and gradients in temperature along the stem (Do and Rocheteau 2002). Further, the effects of variations in wood water content (θ wood) of the stem on the accuracy of TDP measurements have been the subject of investigation (Lu et al. 2004; Tatarinov et al. 2005; Vergeynst et al. 2014). Generally, the depletion and recharge of water storage in stems can lead to substantial fluctuations of θ wood (Nadler et al. 2008; Yang et al. 2015), which may influence wood thermal conductivity (K wood) and subsequently estimates of J s. Based on theoretical analysis of the temperature–θ wood relationship (Carslaw and Jaeger 1959) and a laboratory dehydration experiment on tree stem segments (Vergeynst et al. 2014), it was demonstrated that θ wood influenced K wood around TDP probes and caused underestimations of daytime J s. These underestimations were attributed to selecting one single ΔT max (usually at night) to calculate hourly J s for the whole day (Granier 1987), while ignoring potentially differing K wood between nighttime and daytime. Additionally, the influence of θ wood on ΔT max may differ with soil water conditions, as previous studies found that θ wood in trees and palms fluctuates with θ soil on the longer (i.e., monthly, seasonal) term (Constantz and Murphy 1990; Holbrook et al. 1992; Wullschleger et al. 1996). Further, on rainy days, trunk θ wood was reported to be significantly increased, and subsequently decreased during the following sunny days (Holbrook et al. 1992; Wullschleger et al. 1996; Hao et al. 2013), which may further influence K wood around TDP probes, and thus ΔT max. Ignoring these influences could lead to a potential misinterpretation of the patterns or values of TDP-derived J s.

In a previous study on bamboo water use, underestimated J s by TDP was observed when using the original parameters of the calibration equation (Granier 1985), while newly calibrated, species-specific equation parameters significantly improved the accuracy of the estimation (Mei et al. 2016). Among the potential reasons for the underestimation by the TDP approach on bamboos is the thus-far neglected influence of dynamics in θ wood. Bamboo culms have a large percentage of parenchyma (Liese and Köhl 2015), which provides a potential ‘buffering’ reservoir for transpiration. With the withdrawal from and refilling of water to this reservoir, θ wood may fluctuate accordingly, which can induce changes in culm circumference (Yang et al. 2015). Changes in θ wood in bamboo culms may at least partly be responsible for underestimations of J s by influencing K wood of the culm and consequently ΔT max.

However, the mentioned factors are rather difficult to assess under field conditions and are commonly ignored in TDP studies on bamboos and trees, which is mainly due to practical constraints and the difficulty of measuring the dynamics of temperature around the TDP sensors. One promising approach could be series of controlled numerical simulations of θ wood encompassing different scenarios. Such numerical simulations have previously been successfully applied to investigate the uncertainty of factors such as wood thermal conductivity, non-homogeneity of radial sap flow profiles or external temperature gradients on thermal-based methods including the TDP approach (Tatarinov et al. 2005) and to analyze the influence of wood and probe properties (Wullschleger et al. 2011) and of heat storage capacity (Hölttä et al. 2015) on the accuracy of TDP estimates.

Partially based on such series of numerical simulations, we hypothesized that the change of K wood, responding to diurnal and seasonal fluctuations of θ wood, induces estimation biases in ΔT max and thus in TDP-derived daytime J s; this may (partly) be responsible for the mentioned underestimations of J s. Therefore, the objectives of our study were (1) to test on bamboo segments in a laboratory dehydration experiment whether ΔT max is affected by decreasing θ wood, and to explore if ΔT max in bamboos is influenced by changes in θ soil under field conditions, and (2) to quantify and if necessary correct for potential deviations of J s in bamboo culms with a steady-state thermal model. Our study is intended as a methodological baseline study to evaluate and improve the accuracy of TDP measurements on bamboos.

Methods

Culm θ wood, θ soil and ΔT max

To test if ΔT max is affected by changes in θ wood in bamboos, we applied three different approaches: (1) a dehydration experiment on freshly cut culm segments of Gigantochloa apus, (2) long-term field monitoring of θ soil and daily TDP-derived ΔT max on culms of three bamboo species (Bambusa vulgaris, Dendrocalamus asper, G. apus), and (3) numerical simulation experiments with a steady-state thermal model based on the geometry and physical characteristics of a segment of B. vulgaris.

Laboratory dehydration experiment

Similar to previously conducted dehydration experiments on tree segments (Vergeynst et al. 2014), we performed dehydration experiments on freshly cut culm segments of G. apus; our laboratory experiments took place in May 2013. Before the actual experiments, a freshly sprouted culm of G. apus (diameter 7.3 cm) was cut before sunrise in the common garden of Bogor Agriculture University, Bogor, Indonesia. From the cut culm, three segments (each 20 cm in length) were collected and immediately transported to the laboratory inside a sealed plastic bag to prevent water loss. In the laboratory, the segments were soaked in 40 mM of KCl solution for 24 h to ensure that they reached saturation moisture content. After that, water on the surface of the segments was removed with tissues, while the two ends of each segment were sealed with glue. This ensured that they subsequently only and uniformly dehydrated from the outer culm surfaces.

As a first step of the actual dehydration experiment, the fresh weight of each segment (w fresh, g) was obtained with a balance with 0.01 g resolution (KB2400-2N, KERN & SOHN GmbH, Balingen, Germany). Each segment was then laid down horizontally and a pair of 1 cm-long TDP was installed in the culm wall (Mei et al. 2016). The heating and reference probes were placed 10 cm apart, at 5 cm distance to each end of the segment.

As a second step, cycles of 3-h probe powering and subsequent 2-h dehydration periods were conducted repeatedly over the duration of 5 days. During the powering phase, the heating probe of the TDP sensors was continuously powered with 0.1 W to obtain stable ΔT max readings. During this interval, room temperature was kept constant at about 20 °C and laboratory conditions prevailed (constant light, only little air circulation); the segments thus dehydrated only marginally during this time. During the following 2-h dehydration period, the power of the heating probe was turned off and the segments were placed under an electric fan to artificially accelerate the dehydration process. The segments were further continuously turned to ensure uniform dehydration. At the end of each 2-h period, TDP sensors were removed and the segments were weighted. By continuously repeating the powering-dehydration cycles, data pairs of w fresh vs. ΔT max were produced and recorded.

After the end of the dehydration experiments, the segments were oven dried at 100 °C for 48 h to get their dry weight (w dry, g). With the w dry and w fresh of each powering-dehydration cycle, the θ wood (kg kg−1) was calculated as (w fresh − w dry)/w dry. Subsequently, the relationship between θ wood and ΔT max was examined.

Field monitoring of θ soil and ΔT max

To explore whether, and if so how, ΔT max in bamboo culms was influenced by the θ soil under field conditions, we monitored daily TDP-derived ΔT max on three culms each of D. asper and G. apus and on four culms of B. vulgaris for 7 months (July 2012–April 2013). Simultaneously, θ soil at 20 cm depth was monitored at the respective study sites with time-domain reflectometry sensors (TDR, CS616, Campbell, Logan, USA). For a detailed description of the installation process refer to Mei et al. (2016). Subsequently, the relationship between ΔT max and daily mean θ soil was examined.

θ wood and thermal conductivity for the numerical model

As the theoretical basis of the following numerical model, the relationship of thermal conductivity of wood (K wood) and θ wood was applied following Vandegehuchte and Steppe (2012a, b), who introduced a corrected thermal conductivity for axial directions (K axial, W m−1 K−1):

$${K_{{\text{axial}}}}={K_{\text{w}}}({\theta _{{\text{wood}}}} - {\theta _{{\text{wood-FSP}}}})\frac{{{\rho _{{\text{dry}}}}}}{{{\rho _{\text{w}}}}}+0.04186 \times (21.0 - 20.0 \times {F_{{\text{v-FSP}}}}),$$
(1)

where K w is thermal conductivity of water (0.6 W m−1 K−1), θ wood-FSP is θ wood at the fiber saturation point (%), ρ dry and ρ w are the respective densities of dry wood and water (1000 kg m−3) and F v-FSP is the void fraction of wood at the fiber saturation point. θ wood-FSP and F v-FSP were calculated with several different approaches, using ρ dry and ρ w (see details in Online Resource 1).

To obtain ρ dry of bamboo culms, all culms of the three species that were monitored in our study were harvested at 6:00 am on 15, 16 and 28 April 2013. Segments were obtained every 2 m on the respective culms. The segments were immediately transported to the laboratory in sealed plastic bags. The fresh volumes (v fresh, cm3) of the segments were derived by measuring lengths and inner and outer radiuses of the cylindrical segments; additionally, the w fresh of each segment was established. After that, the segments were dried in an oven at 100 °C for 48 h to get their w dry. Subsequently, ρ d could be calculated as w dry/v fresh.

With the mentioned variables (ρ dry, ρ w, θ wood-FSP and F v-FSP), we calculated series of K axial with θ wood ranging from 0.1 to 1 kg kg−1 (in incremental 0.1 kg kg−1 steps); the thermal conductivity in the transverse direction (K t) was set to half the value of the K axial (Wullschleger et al. 2011). The linear relationship between the K axial and θ wood of the bamboo culms was derived (Fig. 1), and this relationship was applied in the following numerical steady-state thermal model to set the corresponding parameters.

Fig. 1
figure 1

The relationship between the ratio of thermal conductivity in the axial direction (K axial) to culm dry density (ρ d) and culm water content. The relationship was derived by Eq. 1

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Steady-state thermal model

To test if ΔT max decreased with increasing θ wood in bamboos, numerical simulations of temperature distributions were performed with a steady-state thermal model (Academic version, CFX 17.0, ANSYS Inc., PA, USA). The simulations were conducted on a 3D anisotropic grid by numerically solving the steady-state energy balance equation:

$$- \lambda {\nabla ^2}T+{c_{\text{w}}}{Q_{\text{w}}}\nabla T=q,$$
(2)

where q is the heat input of a grid (W m−3), T is the temperature of a grid (K), λ is matrix of thermal conductivity (W m−1 K−1), ∇ is vector differential operator, c w is the specific heat of water (J kg−1 K−1) and Q w is the sap flow vector (kg m−2 s−1). To explore the relationship between θ wood and ΔT max, Q w was set to zero sap flow when aiming to simulate ΔT max (see detailed parameters in Table 1).

Table 1 The parameters for the ANSYS numerical simulation
Full size table

To simplify the simulation, the geometry of the model was based on a simplified 3D bamboo segment, i.e., a cuboid with 20 cm height, 6.65 cm width, and 1 cm depth, ignoring the curvature of the stem surface. The heating probe of the TDP sensor was modeled as an aluminum tube with its actual dimensions, i.e., 0.235 cm in diameter and 1 cm in length (Mei et al. 2016); it was inserted through into the 1-cm-wide simulated culm wall of the cuboid, in the center of the segment (Wullschleger et al. 2011). Resembling the (actual) field methodology, the unheated reference probe was positioned 10 cm upstream from the heating probe. Along the length of the heating probe, the temperature was assumed to be fairly uniform (Wullschleger et al. 2011), and wood physical properties along the probe were also assumed to be uniform. The steady-state simulations were thus simplified for both the front and back surfaces of the segment. Generally, a 2 mm (quadratic) mesh was used for the thermal steady-state model. To better fit the round shape of the heating probe, mesh type was set to quad/tri for the contact area between the heating probe and the surrounding wood.

The boundary conditions of the segment surfaces included inlet, outlet, probe, symmetric surfaces (front and back) and wall. The inlet surface was located on the upstream side and the water came into the segment from the inlet. The outlet surface was located on the downstream side and the pressure was set to 0 Pa. The heating probe was located in the center of the bamboo segment and was powered with 1444 W m−2 (the input power divided by the surface area of the aluminum tube). The front and back surfaces of the wood domain were set as symmetric, which means any plane between the front and back surfaces has same physical and thermal properties. The left and right sides of the bamboo segment were defined as walls with no water flowing out of the segment. The initial temperatures on all segment surfaces and of sap water were set to 300 K (26.85 °C) by default in the ANSYS model.

The influence of θ wood on J s simulated in ANSYS model

To simulate the influence of varying θ wood on TDP-derived J s in bamboo culms, we simulated the ΔT between probes in a series of numerical simulations of J s and θ wood. We incrementally increased J s from 0 to 30 g cm−2 h−1 in 5 g cm−2 h−1 steps and θ wood from 0.1 to 1 kg kg−1 in 0.1 kg kg−1 steps, and thus got a series of ΔT under each combination of J s and θ wood. The ΔT of each possible combination at zero J s was used as ΔT max. The settings of geometry, meshing and change of K a against θ wood followed the previously described model description (see “ θ wood and thermal conductivity for the numerical model”).

Two scenarios were simulated: (1) relative to a fixed nighttime θ wood (θ wood_night, e.g. 1 kg kg−1), where the daytime θ wood (θ wood_daytime) was reduced in 0.1 kg kg−1 steps until its minimum 0.1 kg kg−1; in total, 45 pairs of θ wood_night and θ wood_daytime were simulated; (2) based on different θ wood_night (0.3, 0.6 and 0.9 kg kg−1), where θ wood_daytime was reduced by a constant ratio (i.e., half of θ wood_night). The first scenario was simulated to explore the influence of changes in θ wood between daytime and nighttime (Δθ wood) on diurnal (daytime) J s. The second scenario compared the varying influence of Δθ wood on daytime J s among days with different θ wood_night but with the same relative reduction during the daytime. The second scenario likely occurs, e.g. between different seasons (dry vs. wet season) or among days with different weather conditions (e.g. sunny vs. rainy days).

For a set θ wood_daytime (e.g. θ wood_daytime = 0.1 kg kg−1), the derived ΔT were used to calculate J s in two ways: (1), J s was calculated with ΔT max derived from θ wood_night values equal to the set θ wood_daytime; (2), a biased J s (J s_bias) was calculated with a ΔT max derived from a θ wood_night that was higher than the set θ wood_daytime (e.g. θ wood_night = 0.9 kg kg−1). The respective relative changes of J s (ΔJ s) were calculated as ΔJ s = (J s_bias − J s)/J s. To determine the influence of using different (e.g. biased vs. un-biased) ΔT max on daily accumulated J s, the relationship between ΔJ s and Δθ wood was then applied to one culm of B. vulgaris on a sunny day as a case study.

Correcting J s_bias

To correct the J s_bias, in a first step, we compared ΔJ s to J s_bias for each combination of θ wood_daytime and the mismatched θ wood_night by applying a continuous logistic equation with three variables (ΔJ s_max, ΔJ s_0, r):

$$\Delta {J_{\text{s}}}=\Delta {J_{{\text{s}}\_{\text{max}}}}/\left( {1+\left( {\frac{{\Delta {J_{{\text{s}}\_{\text{max}}}}}}{{\Delta {J_{{\text{s}}\_{\text{0}}}}}} - 1} \right) \times {{\text{e}}^{ - r \times {J_{{\text{s}}\_{\text{bias}}}}}}} \right),$$
(3)

where ΔJ s_max is the maximum ΔJ s; note that as ΔJ s is negative value, ΔJ s_max is the value closest to 0 and means the least relative change between J s_bias and J s; ΔJ s_0 is the ΔJ s when J s_bias is 0; r is the maximum increasing rate of ΔJ s, for each ΔJ s and per unit of increase in J s_bias ((g cm−2 h−1)−1), in analogy to the maximum rate of population increasing per capita per year in logistic regression of population growth (Balakrishnan 1991, vlab.amrita.edu 2011).

As a second step, the three parameters of Eq. 3 (ΔJ s_max, ΔJ s_0, r), for all combinations that passed the logistic regression, were fitted with θ wood_daytime and the mismatched θ wood_night by linear regressions. Then, we replaced the three parameters in Eq. 3 with θ wood_daytime and the mismatched θ wood_night, so that the ΔJ s was instead represented as a function of θ wood_daytime, the mismatched θ wood_night and J s_bias. The thus derived ΔJ s was subsequently used to correct J s_bias:

$$~{J_{{\text{s}}\_{\text{corrected}}}}={J_{{\text{s}}\_{\text{bias}}}}/(1+\Delta {J_{\text{s}}})={J_{{\text{s}}\_{\text{bias}}}}/(1+\Delta {J_{{\text{s}}\_{\text{max}}}}/\left( {1+\left( {\frac{{\Delta {J_{{\text{s}}\_{\text{max}}}}}}{{\Delta {J_{{\text{s}}\_{\text{0}}}}}} - 1} \right) \times {{\text{e}}^{ - r \times {J_{{\text{s}}\_{\text{bias}}}}}}} \right)).$$
(4)

For all data analysis and plotting presented in our study, we used SAS 9.3 (SAS Institute Inc., Cary, NC, USA).

Results

ΔT max and θ wood/θ soil

In the laboratory dehydration experiment, the three freshly sprouted bamboo segments of G. apus showed differences in θ wood vs. ΔT max patterns. Nonetheless, all three segments showed significant negative linear correlations between ΔT max and θ wood (R 2 = 0.56, 0.68 and 0.93 for the segment 1, 2 and 3, respectively; P < 0.05; Fig. 2a). The negative linear relationship was also observed for simulated θ wood and ΔT max derived from the ANSYS model (R 2 = 0.96, P < 0.05; Fig. 2b).

Fig. 2
figure 2

The maximum temperature difference between the probes of TDP (ΔT max) in relation to the water content in culm segments of a freshly sprouted G. apus in a the dehydration experiment and b the ANSYS simulation experiment. Different symbols indicate different segments. The unit of culm water content (kg kg−1) indicates kg water in the culm per kg dry weight

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In the field monitoring, daily mean θ soil was found to have a significant negative linear relationship (P < 0.05) with daily ΔT max for all three bamboo species (D. asper, G. apus, B. vulgaris; Fig. 3). The slope of the ΔT max − θ soil regression line was larger for B. vulgaris (− 3.55) than for D. asper (− 1.91) and G. apus (− 2.14).

Fig. 3
figure 3

The daily maximum temperature difference between the probes of TDP (ΔT max) in relation to daily mean soil moisture for three bamboo species (B. vulgaris: Y = − 3.55X + 10.54, R 2 = 0.63, P < 0.01; D. asper: Y = − 1.91X + 9.21, R 2 = 0.54, P < 0.01; G. apus: Y = − 2.14X + 11.10, R 2 = 0.37, P < 0.01)

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The influence of θ wood on J s

Keeping other controlling variables constant in the ANSYS model, large relative underestimation became apparent (1) for large decreases of θ wood from nighttime to daytime, (2) at relatively low J s, and (3) for relatively larger nighttime θ wood when the ratio of decrease from daytime θ wood was kept constant (e.g. by half).

Using the ANSYS model for series of numerical simulations of θ wood and J s, we found that TDP underestimated daytime J s calculated with nighttime ΔT max when θ wood was lower during the day than during the night. For a given nighttime θ wood (e.g. 1 kg kg−1), lower daytime θ wood (e.g. 0.1 kg kg−1) led to large underestimation of J s of up to 44% (Fig. 4). The ΔJ s (%) was larger at lower J s, and it gradually became smaller and approached to a stable value with increasing J s. For example, the ΔJ s was 18.6% at 5 g cm−2 h−1, while being only 9.4% at 30 g cm−2 h−1, when θ wood was decreased from 1 kg kg−1 (nighttime) by 0.3 kg kg−1 in the daytime (Fig. 4). Even though relative errors were smaller at higher daytime J s, they were responsible for most of the underestimation of daily accumulated J s. Numerical simulations with the ANSYS model for reductions of θ wood_daytime by 0.1 and 0.7 kg kg−1 (corresponding to 0.9 and 0.3 kg kg−1 θ wood_daytime) from 1 kg kg−1 θ wood_night resulted in underestimations of daily accumulated J s by 2 and 19%, respectively (Fig. 5). For example, the relative errors caused at J s over 30 g cm−2 h−1 constituted as much as 64% of the total underestimation of daily water use (Fig. 5).

Fig. 4
figure 4

The simulated relative change of daytime sap flux density (J s) in percentage (%) at different absolute J s calculated with the mismatched ΔT max (J s_bias, g cm−2 h−1). The value on the top of each sub-figure is the night water content (θ wood_night, kg kg−1), and the values at the ends of the lines are daytime water content (θ wood_daytime, kg kg−1). The provided data based on numerical simulations with the ANSYS model

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Fig. 5
figure 5

The corrected sap flux density (J s) for different daytime culm wood water content (θ wood, kg kg−1), a θ wood = 0.9 and b θ wood = 0.3. Simulations based on field monitoring data of a B. vulgaris on 17 September 2012. Numerical simulations with the ANSYS model for daytime θ wood of 0.9, 0.3 kg kg−1 reduced from a 1 kg kg−1 nighttime θ wood result in underestimation of daily accumulated J s by 2, 19%, respectively

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For hypothetical reductions of θ wood_night (0.3, 0.6, 0.9 kg kg−1) to half of their respective values in the daytime (i.e., 0.15, 0.3 and 0.45 kg kg−1), the highest (> 25%) underestimation of daytime J s were simulated for scenarios with high θ wood_night (i.e., 0.9 kg kg−1) under conditions of low J s (e.g. 5 g cm−2 h−1). With increasing J s, the underestimation became smaller (e.g. < 15% at 30 g cm−2 h−1), particularly for lower (i.e., 0.3, 0.6 kg kg−1) θ wood_night (e.g. < 10% at 30 g cm−2 h−1, Fig. 6).

Fig. 6
figure 6

The simulated relative change of daytime sap flux density (J s) in percentage (%) at different absolute J s calculated with the mismatched ΔT max (J s_bias, g cm−2 h−1). Relationships are provided for different nighttime stem water contents (θ wood_night, 0.3, 0.6 and 0.9 kg kg−1), assuming a constant reduction (i.e., by half) in the ratio between nighttime and daytime θ wood. The provided data based on numerical simulations with the ANSYS model

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Correcting J s_bias

Among the 45 combinations of θ wood_night and θ wood_daytime (Fig. 4), there are 21 whose ΔJ s (negative percentage) and J s_bias converged when applying a logistic regression. Two of the three parameters (ΔJ s_max and ΔJ s_0) derived in these 21 logistic regressions were linearly related with both θ wood_night and θ wood_daytime (Eqs. 5, 6), while r was correlated with θ wood_night (Eq. 7), as shown below:

$$\Delta {J_{{\text{s}}\_{\text{max}}}}={\text{ }} - {\text{3}}.{\text{19}}\% - {\text{ }}0.{\text{2}}0\% ~ \times ~{\theta _{{\text{wood}}\_{\text{night}}}}+{\text{ }}0.{\text{25}}\% \times {\theta _{{\text{wood}}\_{\text{daytime}}}} ({R^{\text{2}}}={\text{ }}0.{\text{94}};P<{\text{ }}0.0{\text{1}})$$
(5)
$$\Delta {J_{{\text{s}}\_0}}={\text{ }} - {\text{24}}.{\text{93}}\% - {\text{ }}0.{\text{62}}\% ~ \times ~{\theta _{{\text{wood}}\_{\text{night}}}}+{\text{ }}0.{\text{84}}\% \times {\theta _{{\text{wood}}\_{\text{daytime}}}}({R^{\text{2}}}={\text{ }}0.{\text{91}};P<{\text{ }}0.0{\text{1}})$$
(6)
$$r=0.03\% - 0.00005\% ~\; \times \;~{\theta _{{\text{wood}}\_{\text{night}}}}({R^{\text{2}}}=0.{\text{96}};P<0.0{\text{1}}).$$
(7)

By inserting Eqs. 5–7 into Eq. 4, we derived an equation for correcting the previously discussed estimation biases. With this ‘correction equation’, we thus corrected J s_bias of the remaining 24 combinations of θ wood_night and θ wood_daytime, for which the logistic regression had not worked. Before correction, J s_bias was about 10% smaller than J s (J s_bias = 0.90 × J s − 0.82; R 2 = 0.98; P < 0.01; Online Resource 2 Fig. 2), while corrected J s_bias (J s_corrected) was much closer to actual J s (J s_corrected = 1.01 × J s − 0.64; R 2 = 0.99; P < 0.01; Online Resource 2 Fig. 2).

Discussion

ΔT max and θ wood/θ soil

Granier’s formula for estimating J s with TDP method is based on the assumption that wood thermal properties are constant throughout the day, which results in one constant diurnal ΔT max (Granier 1987). However, ΔT max actually changes as wood thermal properties fluctuate diurnally, which is not reflected when using a common ΔT max. This introduces estimation errors when calculating J s based on a common daily ΔT max (Vergeynst et al. 2014). In our study, a dehydration experiment was conducted on segments of freshly sprouted bamboo culms. The results show that decreasing culm θ wood led to increasing ΔT max (Fig. 2a). Similar results were found in a dehydration experiment on tree segments (Vergeynst et al. 2014). In trees, the fluctuation pattern of θ wood contrasted the daily fluctuation pattern of transpiration. θ wood reached peak values during the night, when J s was zero or marginal, and dropped to a minimum during the daytime (Hao et al. 2013; Sperling et al. 2015). On culms of the bamboo species B. vulgaris, a similar pattern of culm circumference was observed (Yang et al. 2015). Although Yang et al. (2015) did not perform direct measurements of θ wood, the daily dynamics of culm circumference can be expected to at least partly reflect changes in θ wood (Scholz et al. 2008; Köcher et al. 2013). Based on our findings, such fluctuations of θ wood between nighttime and daytime go along with corresponding fluctuations in ΔT max (Fig. 2). In the bamboo culms used in our study, ΔT max significantly decreased with increasing θ soil (Fig. 3), which may be attributed to the corresponding changes of θ wood caused by the dynamics of θ soil. For trees, close coupling of the θ wood–θ soil relationship was reported for rainy days and sunny days and for different seasons; on rainy days or after irrigation, θ wood was significantly increased and subsequently decreased during following sunny days (Holbrook et al. 1992; Wullschleger et al. 1996; Hao et al. 2013). Over the course of a growing season, variation in θ wood of red maple was reported to be 39% between the dry and the wet season (Wullschleger et al. 1996). It has been put forward that dynamics in θ wood may reflect changes in stem water storage (Sperling et al. 2015). Dynamics in stem water storage may thus be derived from changes in θ wood among days, given that θ wood is derived from the ΔT max–θ wood relationship and dry weight or volume of the tree stems or bamboo culms are known.

The influence of θ wood on J s

Using numerical simulations, we found increasing relative underestimations of J s for larger decreases of θ wood from nighttime to daytime (Fig. 4), which may be due to depleted stem water storage. In previous studies on bamboos (Yang et al. 2015) and palms (Sperling et al. 2015), gradual decreases of θ wood from sunrise to sunset were reported. Thus, θ wood in the afternoon and especially at dusk was likely the lowest, which could introduce substantial bias into according estimates of J s (Fig. 4). Our results indicate that the magnitude of the relative underestimation of J s substantially differed with varying J s: the relative error of J s (%) was largest at low J s and gradually became smaller, eventually approaching a stable value with increasing J s (Fig. 4). TDP-derived J s is thus influenced more profoundly by changing θ wood in plants with generally low J s or when J s tends to be low (e.g. early morning, late afternoon). In contrast, for plants with generally high J s or at peak times of J s (e.g. around noon), the influence of changes in θ wood would be smaller. Nevertheless,despite the relatively smaller errors at higher daytime J s, they were responsible for causing most of the absolute underestimations of daily water use (Fig. 5).

Additional to varying for different J s, the influence of θ wood on TDP-derived J s may also differ with different water conditions of the soil due to different weather or management conditions, e.g. between rainy or sunny days or after irrigation. The applied model simulation indicated that, when θ wood was reduced by the same ratio (e.g. half) from nighttime to daytime, higher nighttime θ wood caused larger relative underestimation of J s (Fig. 6). High θ wood_ night may occur during the wet season or during the growing period of a plant. This could potentially lead to estimation errors when calibrating the TDP method with other methods in situ. Consequently, calibration approaches conducted over short periods may not reflect medium- or long-term dynamics in θ wood. Similarly, one-time laboratory calibration experiments on tree or bamboo segments may also be prone to error due to potentially varying (or unknown) θ wood, e.g. because of varying soil water conditions at the time of stem harvest.

Correcting J s_bias

In our study, we numerically simulated the influence of θ wood on TDP-derived J s. The results point to a direct and quantifiable relationship between θ wood and relative changes in J s, and thus potential errors in previous studies when assuming a static θ wood (Wullschleger et al. 2011; Vergeynst et al. 2014). Wullschleger et al. (2011) simulated scenarios examining the impact of K wood on the relationship between J s and k (= ΔT max/ΔT −1), and they pointed out that the J s–k relationship might be influenced by several factors including θ wood, wood density, and K wood. Vergeynst et al. (2014) quantified the influence of θ wood on J s by simulating possible temperature changes that were assumed to relate to θ wood. Our results supplement these previous TDP error analyses by building a direct and quantifiable relationship between θ wood and J s_bias with a numerical model. In our study, J s_bias successfully corrected with our derived ‘correction equations’ (Eqs. 4–7). The four equations (Eqs. 4–7), which built upon 21 of the 45 sets of the simulated data, yielded satisfactory corrections of J s_bias when applying them to the remaining 24 (Online Resource 2 Fig. 2). Nonetheless, our correction approach still needs further improvements for future applicability.

First, in our study, the model was built under steady-state conditions, under which each simulation result was derived from the assumption of constant J s. These steady-state conditions may not (always) be met under field conditions, where J s is prone to external influences and may thus change frequently and not always predictably. For future studies, a non-steady-state with varying J s should be considered, and an application of the model in situ for simultaneous assessments of θ wood and J s would be needed. Additionally, the simulation in our study was mainly based on Eq. 1, derived by applying density and porosity of the bamboo species B. vulgaris. For other (bamboo) species, the parameters in the equations may thus differ. Further correction attempts could for e.g. directly include the density and porosity of wood into the correction equations as further variables, making it more universally applicable. In addition, it has to be considered that nighttime sap flow may occur in some species and that ‘zero sap flow’ conditions might thus not always be met. Our model-derived ΔT max–θ wood relationship was based on the assumption that the nighttime zero sap flow existed and lasted long enough to establish a one-to-one correspondence between ΔT max and maximum θ wood. This ideal assumption may not always be met during field experiments (Regalado and Ritter 2007). As such, several studies have reported nighttime sap flow in different species, which was presumably related to atmospheric evaporative demand (Forster 2014). Further, during dry periods, the commonly high daytime transpiration in combination with relatively low soil–water availability could potentially lead to more nighttime sap flow in the form of refilling depleted stem water reservoirs (Wang et al. 2012). In some cases, this might mean that zero flow conditions are not met at all, not even during the night. Even when the stem is fully recharged and reaches its maximum water content, nighttime sap flow could still occur, e.g. in the form of guttation. In this case, the ΔT max might be derived under a non-zero sap flux conditions and might lead to underestimation of J s. Therefore, both changes in θ wood and nighttime sap flow are likely to influence ΔT max in the same direction. It will require further, more in-depth ecophysiological studies to evaluate and correct the impact of nighttime sap flow as well as the coupled effects of both changes in θ wood and nighttime sap flow on TDP-derived J s.

For TDP and other heat-based methods, influences or biases due to changes in θ wood cannot fully be avoided (Vergeynst et al. 2014). Decreasing the sensitivity of the heat domain by increasing the power supply (Tatarinov et al. 2005) may be an option. However, such operation may lead to possible damages to the wood structure. Additionally to calibration-based and mathematical approaches to correct for such errors due to changes in θ wood, other possible solutions, as previously explored by Vandegehuchte and Steppe (2012b) and Trcala and Čermák (2016), may include new types of sensors that already account for dynamic changes of θ wood when estimating J s. The model applied in our study to simulate the influence of changes in θ wood on J s may be a possible reference for future studies to develop new models and for testing and further improving such new types of sensors.

Conclusions

In our study encompassing laboratory dehydration experiments and field monitoring, ΔT max, a core variable in calculating J s with the TDP method, was found to correlate negatively with both θ wood in bamboo culms and with θ soil. By numerically simulating this negative ΔT max–θ wood relationship for different scenarios of daily and seasonal changes in θ wood, the corresponding relative underestimation of J s was quantified. Keeping other controlling variables constant, large relative underestimation became apparent (1) for large decreases of θ wood from nighttime to daytime, (2) at relatively low J s, and (3) for relatively larger nighttime θ wood when the ratio of decrease to the daytime (e.g. by half) was kept constant. Our findings indicate that TDP measurements can be profoundly influenced by diurnal changes in θ wood, particularly in species with low water consumption, in species with large diurnal changes in stem water storage (between nighttime and daytime), and between periods with strongly alternating soil–water conditions (e.g. between sunny and rainy days). A mathematical correction equation was built with the simulated data by the steady-state numerical model, and it yielded acceptable corrected J s. Interesting approaches for future studies include testing the here applied model in situ by simultaneously assessing dynamics in θ wood and J s, as well as further improving and developing heat-based methods to include the assumption of non-stable θ wood at different temporal scales.

Author contribution statement

TM designed the experiment; TM, DF, AR performed measurements; DF developed the model and analyzed the data; TM, DF, DH and AR wrote and revised the paper.